7 


jpLxJd^  ^        ^     ^    ' 


y-^^^^.^  9f  I  J  fX, 


ELEMENTS   OF 

PLANE   AND   SPHERICAL 

TRIGONOMETRY 


■y^y^ 


THE  MACMILLAN  COMPANY 

NEW  YORK   •    BOSTON    ■    CHICAGO 
ATLANTA  •    SAN    FRANCISCO 

MACMILLAN  &  CO.,  Limited 

LONDON  •  BOMBAY  •  CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  Ltd. 

TORONTO 


ELEMENTS    OF 

PLANE    AND    SPHERICAL 

TRIGONOMETRY 


BY 


DAVID   A.    ROTHROCK,  Ph.D. 

PROFESSOR    OF    MATHEMATICS,    INDIANA    UNIVERSITY 
BLOOMINGTON,    INDIANA 


THE   MACMILLAN   COMPANY 
1911 

All  rights  reserved 


COPYRIfillT,    1910, 

Br  THE   MACMILLAN  COMPANY. 


Set  up  and  electrotyped.     Published  September,  igio.     Reprinted 
January,  1911. 


NoTtoootr  Jfittea 

J.  8.  Gushing  Co.  —  Berwick  &  Smith  Co. 

Norwood,  Mass.,  U.S.A. 


Engineering  & 

Mathematical 

Sciences 

Library 

5-31 


m 


PREFACE 

In  this  work  the  author  has  endeavored  to  prepare  a  text 
which  would  serve  as  a  basis  for  a  fifty-  or  sixty-hour  course 
in  Plane  and  Spherical  Trigonometry  as  ordinarily  presented 
in  advanced  secondary  and  elementary  college  courses. 

Emphasis  is  placed  upon  drill  work   in  the  trigonometric 

identities,  upon  the  applications  of  trigonometry  to  practical 

problems,    and   upon   approximate   calculations   by   means    of 

\»    natural  functions-      The   more   accurate   results   obtained   by 

y\^    logarithmic    calculations    are     emphasized    in    the    solutions 

^    of  oblique  triangles ;  a  uniform  style  of  tabulating  logarithmic 

X     calculations  is  suggested. 

^         For  the  benefit  of  those  who  may  wish  to  pursue  advanced 

^    courses  in  mathematics,  a  brief  discussion  of  analytic  trigo- 

<"w    nometry  is  presented  in  Chapter  IX.     In  Part  II  the  elements 

of  spherical  trigonometry  are  developed  in  so  far  as  to  include 

the  ordinary  formulae  necessary  in  the  solution  of  right  and 

oblique  spherical  triangles. 

DAVID   A.   ROTHROCK. 
Bloomington,  Indiana, 
December,  1909. 


374041 


Digitized  by  the  Internet  Arcinive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/elementsofplanesOOrothiala 


CONTENTS 

PART   I 
PLANE  trigo:n^ometry 

CHAPTER   I 
Trigonometric  Functions  of  Acute  Angles 

ART.                                                                                                                                                                                •  PAG  8 

1.  Trigonometry 1 

2.  Functions  of  Acute  Angles 2 

3.  Functions  of  Complementary  Angles 3 

Exercises         ....         ....'...  5 

4.  Fundamental  Relations  among  the  Trigonometric  Functions         .        .  6 

5.  Fundamental  Identities 7 

6.  Variation  of  the  Trigonometric  Functions     .                  ....  7 

7 .  Transformation  of  Identities  . 7 

Exercises 8 

Exercises 10 

8.  Functions  of  Particular  Angles :  0°,  30°,  45°,  60°,  90°  .         ...  13 

Exercises 14 

9.  Table  of  Trigonometric  Functions 15 

CHAPTER   II 
Solution  of  Right  Triangles 

10.  Fundamental  Formulas .        . 16 

11.  Projections 17 

12.  Components 18 

13.',  Resultant 18 

14.  Projected  Areas 21 

15.  Areas  of  Right  Triangles 21 

16.  The  Isosceles  Triangle 22 

17.  Notations  of  Direction   .         .         . 23 

18.  To  solve  Right  Triangles 25 

Exercises 26 

CHAPTER   III 
Trigonometric  Functions  of  Any  Angle 

19.  Axes.     Quadrants          . 31 

20.  Coordinates,  Abscissa,  Ordinate 31 

21.  Definitions  of  the  Functions 32 

vii 


viii  CONTENTS 


22.  Laws  of  Signs 33 

Exercises 34 

23.  Functions  of  Negative  Angles 34 

Exercises 35 

24.  Functions  of  90°  -  tf  and  90°  +  tf 36 

25.  Functions  of  180'"  -  0  and  180°  +  6 37 

Exercises 38 

26.  Line  Values  of  the  Functions 39 

Exercises 41 

27.  Graphs  of  the  Trigonometric  Functions 42 

CHAPTER   IV 
Measurement  of  Angles 

28.  L'^nits  of  Measure 44 

29.  Relations  between  Degree  and  Radian  Measure 44 

Exercises 45 

30.  The  Length  of  any  Arc 46 

31.  Segment  and  Sector  Areas 47 

Exercises 47 

CHAPTER  V 
Functions  of  Two  Angles.    Multiple  Angles 

32.  To  develop  sin  (« +  /3)  and  cos  («+/3) 50 

33.  To  develop  tan  (a  +  )3) 51 

34.  Important  Formulas 52 

Exercises 52 

Functions  of  Multiple  and  Sub-Multiple  Angles 

35.  Functions  of  2  « 53 

36.  Functions  of  3  a 54 

37.  Half-angle  Formulas      .        .        .         .  • 54 

Exercises 55 

Sum  and  Difference  Formulas 

38.  Converting  to  Products 56 

39.  Converting  to  Sum  or  Difference 67 

Exercises 57 

CHAPTER  VI 
Logarithms 

40.  The  Index  Laws 60 

41.  Definition  of  Logarithms 61 


CONTEXTS 


IX 


42.  Systems  of  Logarithms 

43.  Laws  governing  the  Use  of  Logarithms 

44.  Characteristic  and  Mantissa  .... 

45.  Use  of  Tables 

46.  Conversion  of  Coipmon  to  Napierian  Logarithms 

Exercises  in  Use  of  Logarithms 


62 

62 
63 
66 
67 
67 


CHAPTER  VII 
Solution  of  Triangles  in  General 

47.  The  Theorem  of  Sines 

Applications  of  the  Theorem  of  Sines     , 

48.  The  Theorem  of  Tangents      .... 

Applications  of  the  Theorem  of  Tangents 

49.  The  Theorem  of  Cosines        .... 

Exercises  ...... 

60.  The  Half-angle  Theorems       .... 

Application  of  the  Half-angle  Theorems 

Areas  of  Triangles 

51.  Area  in  Terms  of  Sides  and  Angles 

52.  Area  in  Terms  of  r 

63.    Expressions  for  Area     ..... 

Exercises 


71 
73 

74 
76 
77 
77 
79 


80 
81 
82 
82 


CHAPTER   VIII 
Inverse  Functions.    Trigonometric  Equations 

54.  Inverse  Notation 87 

55.  Inverse  Identities  .         ..........  88 

Exercises 89 

56.  Definitions 90 

57.  Solutions 91 

58.  Simple  Equations 91 

Exercises 92 

59.  Equations  of  the  Form  r  cos  0  =  a,  r  sin  0  =  6       .....  93 

60.  Equations  of  the  Form  r  sin  d  cos  (p  =  a,  r  sin  ^  sin  0  =  6,  r  cos  d  =  c       .  94 

61.  To  solve  a  •  sin  a;  +  ft  •  cos  x  =  c 96 

62.  To  solve  sin  {x  -f  <^)  =  a  sin  x .95 

63.  To  solve  tan  {x  +  0)  =  a  tan  x 96 

64.  To  solve  x  =  a  +  ^s\vix 97 


CHAPTER   IX 

Complex  Numbers.    DeMoivre's  Theorem.     Trigonometric  Series. 

Exponential  and  Hyperbolic  Functions 

65.    Roots  of  Quadratic  Equations 

Exercises 


100 


X  CONTENTS 

ART.  PAGB 

66.  Complex  Numbers  expressed  Trigonometrically 100 

67.  DeMoivre's  Theorem 101 

68.  Raising  to  Powers  and  extracting  Roots 102 

69.  Value  of  sin  at,  cos  x  in  Terms  of  a; 104 

70.  Summation  of  Series 108 

Examples 109 

71.  The  Exponential  Series 110 

72.  Euler's  Formulas Ill 

Exercises 112 

73.  The  Hyperbolic  Functions 112 

74.  The  Gudermannian 116 


PART   II 
SPHERICAL   TRIGONOMETRY 

CHAPTER   X 

General  Definitions.     The  Right  Triangle 

75.  Definitions  and  Geometric  Properties 117 

76.  The  Polar  Triangle 118 

The  Right  Spherical  Triangle 

77.  Definitions 119 

78.  Trigonometric  Relations 119 

79.  Important  Formulas 120 

80.  Napier's  Rules  of  Circular  Parts 121 

81.  Relative  Dimensions  of  Sides  and  Angles 122 

82.  The  Isosceles  and  Quadrantal  Triangles 122 

83.  Solution  of  Right  Spherical  Triangles 123 

Exercises 124 


CHAPTER   XI 

The  Oblique  Spherical  Triangle 

84.  The  Theorem  of  Sines 

85.  The  Theorem  of  Cosines.     Side  and  One  Angle 

Modified  Formula   ..... 

86.  The  Theorem  of  Cosines.     Angles  and  One  Side 

87.  The  Half-angle  Formulae        .... 

1)  The  Half-angles  in  Terms  of  the  Sides  . 

2)  The  Half-sides  in  Terms  of  the  Angles 

88.  Napier's  Analogies 

89.  The  Area  of  a  Spherical  Triangle  . 

90.  Solution  of  Oblique  Sp}ierical  Triangles 

Exercises 


126 
127 
128 
128 
129 
129 
130 
131 
133 
133 
135 


CONTEXTS  xi 
Applications  of  Spherical  Trigonometry 

ART.  PAOM 

91.  The  Earth  as  a  Sphere.     Definitions  and  Notation        ....  137 

92.  The  Terrestrial  Triangle 138 

93.  The  Celestial  Sphere 140 

94.  The  Celestial  Triangle 141 

Exercises 142 


CHAPTER   XII 
Formulas 143 


TRIGO^^OMETRY 

PART    I 
PLANE   TRIGONOMETRY 

CHAPTER   I 
TRIGONOMETRIC   FUNCTIONS   OF   ACUTE   ANGLES 

1.    Trigonometry.     The  word    Trigonometry  is  derived  from 

two  Greek  words,  triangle  (jpi^ycovov^  and  measurement  (^fierpov), 
which  would  suggest  that  the  subject  has  to  do  with  the  meas- 
urement of  the  triangle.  At  the  present  time  the  subject  of 
trigonometry,  though  treating  of  the  measurement  of  the  tri- 
angle, has  a  much  wider  scope,  and  includes  all  manner  of 
investigations  depending  upon  certain  functions  of  angles 
called  Trigonometric  Functions.  These  functions  are  called 
sine,  cosine,  tangent,  cotangent,  secant,  and  cosecant.  For  any 
angle  6  *  these  words  are  abbreviated  into  sin  6,  cos  6,  tan  6,  cot  6, 
sec  6,  CSC  6,  respectively.  In  the  next  section  definitions  are 
given  for  the  trigonometric  functions  of  an  acute  angle,  but  it 
should  be  borne  in  mind  that  similar  definitions  are  applicable 
for  any  angle  whatever,  and  later  will  be  extended  to  angles 
varying  from  0°  to  360°. 

*  In  trigonometric  notation,  angles  are  frequently  denoted  by  Greek  letters. 


The  Greek  alpliabet  is 

here  inserted. 

Letters          Names 

Letters 

Names 

A  a         Alpha 

I  I 

Iota 

B  /3        Beta 

Kk 

Kappa 

r  7         Gamma 

AX 

Lambda 

A  S         Delta 

Mm 

Mu 

E  £         Epsilon 

N  i> 

Nu 

Z  f        Zeta 

Sf 

Xi 

H  T7        Eta 

0  0 

Oniicron 

e  e        Theta 

n  IT  u> 

Pi 

B 

1 

Letters 

Names 

Pp 

Rho 

S  <rs 

Sigma 

T   T 

Tau 

T  i; 

Upsilon 

#  <t> 

Phi 

Xx 

Chi 

-ir  f 

Psi 

a  w 

Omega 

PLANE   TRIGONOMETRY 


[§2 


2.    The  Trigonometric  Functions  of  an  Acute  Angle. 

(1)  Definitions.  For  an  acute  angle  6  constructed  in  a  right 
triangle,  Fig.  1,  with  x,  y,  r,  as  the  base,  altitude,  and  hypote- 
nuse, respectively,  we  define 


Fig.  1. 


.    A     opposite  side     y 

sin  D  =  —c 1 =  — » 

hypotenuse      r 

A     adjacent  side     x 

cos  0  =  -ir — i =  —  J 

liypotennse       r 

Q     opposite  side  _  y 
~  adjacent  side  "  a; ' 


liypotennse  _     1 
opposite  side  ~  sin  6 

liypotennse  _     1 
adjacent  side  ~  cos  6 

,  n     adjacent  side        1 
cot  9  =  -^ — iT — vx-  =  ^ — n 
opposite  side     tan  6 


CSC  6 


sec  6 


r 

r 

r 

'  x' 

oc 
V 


To  these  six  functions  are  sometimes  added : 

versed  sine  9  =  1  —  cos  9,  written  vers  9, 
coTersed  sine  9  =1  —  sin  9,  written  corers  9. 

These  six  functions  of  the  angle  6  are  called  trigonometric 
functions  (trigonometric  ratios)  of  that  angle.  The  symbols 
sin  ^,  cos  6,  tan  ^,  cot  ^,  sec  6,  esc  ^,  are  usually  read  sine  0, 
cosine  0,  tangent  0,  cotangent  0,  secant  0,  cosecant  0^  respectively. 

(2)  Elementary  relations  among  the  functions.  The  six  func- 
tions sin  ^,  cos  ^,  tan  ^,  cot  ^,  sec  ^,  esc  0^  are  not  independent. 
By  comparing  the  definitions  we  find  : 

(1)   tan  9  =  -^,  or  tan  9  cot  9  =  1, 
^  ^  cot9  ' 


(2)   c«s9  = 


sec  9 


or  cos  9  •  sec  9  =  1, 


(3)   sin  9 


csc9 


or  sin  9  •  CSC  9  =  1, 


§2]      TRIGONOMETRIC   FUNCTIONS  OF   ACUTE   ANGLES 


(3)  Exponents.  When  the  trigonometric  functions  are  to  be 
affected  by  exponents,  the  following  notation  is  usually  em- 
ployed :  (1)  if  the  exponent  be  positive,  the  index  is  placed 
thus,  sin^^,  tan^^,  sec"^,  and  read  sine  square  6,  etc. ;  (2)  if 
the  exponent  be  negative,  the  index  is  usually  attached  to  a 
bracket  as  in  bracketed  expressions  in  algebra,  thus  (sin  6}~^  = 

^,    (Got0Y^= — r-r,    (cos  ^)~"= ;:.      Tlicsc   arc   read 

sin^     ^  ^         cot2  6'     ^  ^         cos»^ 

sin  6  expo7ient  minus  one.,  cot  6  exponent  minus  two,  cos  6  expo- 
nent mhius  n. 

(4)  Functions  are  constant  for  any  given  angle.  The  trigo- 
nometric functions  of  an  acute  angle  are  ratios  of  lines  in  a 
right  triangle.  They  are  constant  for  any  fixed  angle,  and  do 
not  change  value  for  different  lengths  of  the  sides  of  the  triangle. 
Thus,  Fig.  2, 

C" 
B' 


Sin  a  =  ——  = 


cos  a  =  -—-  = 


tan  a  =  -— -  = 


BC     B'O'     B"0" 


OB      OB' 

OB"' 

00     oc 

OC" 

OB      OB' 

OB"' 

BC     B'C 

B"C" 

00       00' 


oc 


Tables  of  these  ratios  have  been  constructed,  showing  their 
numerical  values  for  all  angles  from  0°  to  90°  (see  Table  p.  15). 

3.    Functions  of  Complementary  Angles.     In  Fig.  3,  6  and  ^ 
are  complementary  angles  :  6  -\-  (f>  =  90°. 


PLANE   TRIGONOMETRY 


[§3 


sin  e  =  ^  =  cos  <|), 
r 

CSC  8  =  -  =  sec  (j), 

cos  e  =  ^  =  sin  4», 

sec  6  =  --  =  CSC  <j), 

tane  =  ^  =  cot<t), 

cot  e  =  ^  =  tan  (j). 

Fig.  3. 

The  cosine,  cotangent,  and  cosecant  of  an  angle  are  co- 
functions  of  the  sine,  tangent,  and  secant,  respectively. 

From  the  definitions  we  see  that  any  function  of  an  angle 
equals  the  co-function  of  the  complement  of  that  angle. 

For  example,  sin  30°  =  cos  60°,  tan  35°  =  cot  55°,  esc  32°  =  sec  58°, 

sin  (90°  -A)=  cos  A,  esc  (90°  -  A)  =  sec  A, 

cos  (90°  -A)  =  sin  A,  sec  (90°  -  A)  =  esc  A, 

tan  (90°  -A)  =  cot  A,  cot  (90°-  A)  =  tan  A. 

Examples.  Fill  the  blanks  in  the  following  with  the  proper 
co-function : 


1.  sin  48°=    i>*  *<  ^ 

2.  sin  84°  = 

3.  tan  25°  48'=     ^M 

4.  cot  87°  50'  = 

5.  cos  41°  50'  = 

6.  secl0°15'  = 


7.  cos  (99°- a:)  = 

8.  tan  (90°  -  10°)  = 

9.  sin  (90°  -  100°)  = 

10.  cot  (80°  -  x)  = 

11.  CSC  67°  10'  = 

12.  tan  (90°  -x  +  y')  = 


§3]      TRIGONOMETRIC   FUNCTIONS  OF   ACUTE   ANGLES 


EXERCISES 


1.  Construct  an  acute  angle  a  sucli  that  sin  «  =  f ,  and  write 
the  remaining  trigonometric  functions. 

Solution,  sin  «  =  f  tells  us  that  the  side 
of  the  triangle  opposite  a  is  3,  and  the 
hypotenuse  is  5 ;  hence,  the  construction  is 
as  shown.  "Writing  the  functions  of  a  from 
the  drawing,  we  have 

sin  «  =  f,      cos  «  =  |, 

tan  «  =  I,      cot  «  =  |, 

sec  a  =  f,      esc  a  =  |. 

2.  Construct  a  when  tan  a  =  |,  and  write  the  other  functions 
of  a.     Write  the  functions  of  90°  —  a. 


y 

V 

/' 

5 

/^ 

/^ 

V 

< 

cr 

^ 

\ 

4 

( 

« 

) 

1 

y 

^ 

<< 

0 

\y 

^ 

>1 

/' 

€ 

/\ 

\\ 

3 

(«» 

A_ 

Solution,     a    is  constructed    as   shown 
in  fiffure. 


sina= -,   " 

cos  a  = , 

V13 

V13 

o 
tan  a  =  " , 
3 

cota  =  -, 

seca=^l^^ 

csc«-^^, 

2 

90°-a  =  /3;  sin/3  = 

3 

=  cos  a  = ,  etc 

\/13 

Construct  the  angle  a  in  the  following  cases  and  write  the 
remaining  functions: 


3. 

sin«=iV 

4. 

cos  «  =  |. 

9. 

sin  «  =  ^. 

10. 

tan  «  =  |. 

11. 

sin  «  =  -^j. 

12. 

tan  «  =  2. 

13. 

cot  a  =  3. 

14. 

sin  a  =  1- V3 

15. 

tan  a  =  lV8 

5. 

cot  a  =  |. 

7.    tan  a  =  f . 

6. 

sec  a  =  |. 

8.    cot  a  =  ^§g. 

16. 

sec  a  =  4. 

17. 

sin  a  =  0.6. 

18.      COS  « 


19.    tan  a 


z  ni 
1  +  m^ 

Vl  —  w 


wi 


3  a: 
20.    cot  a  = 

4 


PLANE   TRIGONOMETRY 


[§4 


In  the  right  triangle  ABG^  Fig.  4,  if 

21.  sin  A  =  |,  c  =  24,  find  a. 

22.  sin  A=\^  c  =  24,  find  a. 

23.  tan  J.  =  |,  6  =  16,  find  a. 

24.  tan  vl  =  |,  c  =  15,  find  5. 

25.  tan  J.  =  0.6,  a  =  10,  find  h. 

26.  sin  yl  =  |,  a  =  15,  find  c. 

27.  COS  ^  =  f ,  ^  =  21,  find  h  and  a. 

28.  sec  vi  =  5,  6  =  50,  find  e  and  a. 

29.  tan  A  =  0.25,  J  =  40,  find  the  area. 

30.  CSC  ^  =  2.5,  a  =  100,  find  c  and  the  area. 

4.    Fundamental  Relations  among  the  Trigonometric  Functions. 
Let  6  denote  any  acute  angle,  Fig.  5.     From  plane  geometry, 

a^  +  y2^r2.  (1) 

Defining  sin  6  and  cos  ^, 

qj  q^ 

sin  6  =  -,  cos  ^  =  -, 
r  r 

we  have,  on  squaring  and  adding. 


Fig.  5. 


sin2  6'+cos2^  =  ^'^4^=l.     (A) 


From  the  same  figure,   sec  0=  -,  tan  0  = 


y 


Squaring  and  subtracting, 

sec2  e  -  tan2  B  = 


x" 


=  1,  from(l).  (5) 


Combining  esc  6  and  cot  ^  in  a  similar  manner, 

K.2 


C8C2  ^  -  C0t2  ^  =  ^^^-^  =  1. 


Also, 


tan  <9  =  ^  = 

X         X 


y  _y^r 

-=-  r 

cos  6 


r 

sin  0 
cos^' 


X 


sec  /9  =  - 

X 


cot^  =  -  =  -^--, 
z/      sin  a 

1 


cos  d' 


(F} 


CSC  ^  =  -  =  -: ri. 

y      sin  V 


,o 


sJi 


CE) 

cota  =  :, = 

tana 

cos  a 
sma 

(F) 

sec  a  = » 

cos  a 

1 
esc  a  =   . 

r^f-- 


§§5-7]    TRKiONOMETRIC  FUNCTIONS  OF  ACUTE  ANGLES        7 

As  fundamental  relations  among  the  six  functions,  we  may 
enumerate  formulas  A,  B^  C,  above,  and  the  identities  coming 
directly  from  the  detinitions  in  §  2. 

5.  Fundamental  Identities.  Collecting  important  results,  we 
have,  for  any  angle  a : 

(A)  sin2a  +  cos-a=  1, 

(B)  sec-  a  -  tan-  a  =  1, 

(C)  csc2  a  -  cot-  a  =  1, 

sin  a 

(D)  tana  = > 

cos  a  sin  a 

These  identities  should  be  memorized. 

6.  Variation  of  the  Trigonometric  Functions.  It  should  be 
noticed  that  as  the  angle  increases  from  0°  to  90°,  the  functions 
vary  as  follows : 

(1)  si7ie  increases  from  0  ^o  1, 

(2)  cosine  decreases  from  1  to  0, 

(3)  tangent  increases  from  0  to  cc  , 

(4)  cotangent  decreases  from  oo  to  0, 

(5)  secant  increases  from  1  to  co  ^ 

(6)  cosecant  decreases  from  ob  to  1. 

These  facts  may  be  observed  from  the  above  identities,  as 
well  as  from  the  definitions  of  the  functions  in  §  2. 

7.  Transformation  of  Identities.  By  means  of  the  funda- 
mental identities,  §  5,  any  trigonometric  function  may  be 
changed  into  various  forms. 

For  example, 

•      ,         /I 9— r     ^  f-t  i  Vsec^  <f>  —  1      tan  <b      . 

sm  (p  =  Vl  —  cos-^  (^  =  -v/1  —  ■ — -—  = ^ = ^,  etc. 

^        sec^  <j)  sec  (j)  sec  <f) 

These  forms  for  the  value  of  the  sine  of  an  angle  are  ob- 
tained by  using  identities  (-4),  (j^),  (-B),  and  algebraic  manipu- 
lation. 

Again, 

sin  a  •  sec  a  •  cot  a  = •  cot  a  =  tan  a  •  cot  a  =  1. 

cos  a 


8  PLANE   TRIGONOMETRY  [§  7 

In  verifying  an  identity,  we  may  proceed,  (a)  by  transforming 
the  left  member  of  the  equation  into  the  right,  (5)  by  transforming 
the  right  into  the  left,  or  (c)  by  reducing  each  side  to  the  same 
form. 

For  example,  show  cos*  a:  —  sin^ar  =  2  cos^x  —  1. 

(rt)  Transform  the  left  member  into  the  right. 

cos^x  —  sin*x  =  (cos"^a:  +  sin"^  x)  (cos^  a;  —  sin^z) 

=  cos2  X  —  sin^  X  by  (.4  ) 

=  cos^x  —  (1  —  cos'^x)  by  (J) 

=  2  cos'^x  —  1. 

(ft)  Transform  the  right  member  to  the  form  of  the  left. 

2  cos^x  —  1  =  cos^x  —  (1  —  cos^x) 

=  cos"^x  —  sin^x  by  (J) 

=  (cos'^x  —  sin2x)(cos2x  +  sin^x),  multiplying  by  1, 
=  cos*  X  —  sin*  X. 

(c)  Transform  each  side  to  a  common  form. 

cos*  X  —  sin*  X  =  2  cos^  x  —  1 

=  cos-^x  —  sin^ar. 

Remove  the  factor  cos^x  4  sin^x  =  1  from  the  left  member,  and  we  have 
cos^x  —  sin^x  =  cos^x  —  siu^a:. 

The  student  should  attain  a  good  degree  of  skill  in  manipu- 
lating trigonometric  functions  ;  to  this  end  is  now  inserted  a  list 
of  the  more  common  forms  of  identities  involving  a  single  angle. 

EXERCISES 

Verify  the  following  identities  : 

1.  cos2^(l  +  tan2^)  =  l. 

2.  (sec2  A  -  1)  (csc2  A-\}^\ 

3.  tan  A  +  cot  A  =  sec  A  x  esc  A. 

4.  sin^  J.  (csc^^  —  1)  =  cos^^. 

5.  cot^  A  —  cos^  A  =  cos^  A  cot^  A. 

cos  A  .A 

6.  - — — —  =  tan  A. 

sin  A  cot^  A 

7.  sin  A  cos  A  (tan  A  +  cot  A)  =  1. 


§7]      TRIGONOMETRIC   FUNCTIONS   OF   ACUTE   ANGLES         9 

8.  (tan  A  +  cot  Ay  =  sec2  A  +  csc2  A. 

9.  cos  yl  CSC  ^  tan  ^  =  1. 

10.  COS  A  (cot  A  +  tan  A)  =  esc  A. 

,-.     ^.4^    4   ,       sin  J.  . 

11.  cot  A  H =  CSC  A. 

1  -f-  cos  A 
,,„     sec  A     tan -4     ^ 

^12. -  =  1. 

cos  A      cot  A 

13.  sec^^J.  — sec2^sin2^  =  l. 

14.  (CSC  A  -  cot  AY=^~  ^"''•'^  ^. 
^  ^       1  +  cos  A 

15.  sec  yl  —  cos  yl  =  sin  A  tan  ^. 

sin  A  ,  cos  ^      1 

16.  — 1 =  1. 

CSC  A     sec  A 


17. 


18. 


tan  A  —  1  _  1  —  cot  ^ 
tan  A  +  1  1  +  cot  ^ 
sec  A  —  CSC  yl      tan  A  —  1 


sec  ^  +  CSC  A      tan  A  +  1 

19.  cos^^-sin*vl  =  cos2^-sin2vl  =  l-2sin2^=2cos2J.— 1. 

^     1  +  cot^  ^  ,„   , 

20.  -— i^ — —  =  cot^  A. 

1  + tan2  A 

21.  (sin  ^  +  cos  Ay  +  (sin  A  —  cos  J.)^  =  2. 

22.  sec  ^  +  tan  A  =  (sec  ^  —  tan  ^1)"^. 

23.  sin*  A  +  cos*  A  =1  —  2  sin^  A  cos^  ^, 

^    24.    sin^^4-cos^^=  (sin  J.4- cos  J.)(l  — sin^  cos-4.). 

25.  sec*  .-L  —  tan*  A  =  sec^  ^  +  tan^  A  =  l+2  tan^  J.. 

26.  *    Show  3^ -i- 1/^  =  1^,  when  x  =  r  •  cos  ^,  y  =  r  •  sin  ^. 

27.  Show  —  +  4;  =  1,  when  a;  =  a  •  cos  ^,  y  =  h  ■  sin  ^. 

28.  Show  —  —  ^  =  1,  when  a;  =  a  •  sec  <^.    y  =  h  ■  tan  d). 

29.  Show  a:^  _(_  ^2  ^  2^  =  r^,  when  x  =  r  •  sin  ^  •  cos  <^, 

y  =  r  •  sin  ^  •  sin  <^,  z  =  r  -  cos  ^. 

*  The  algebraic  equations  in  Exercises  26-29  are,  respectively,  the  equations 
of  a  circle,  ellipse,  hyperbola,  and  sphere.  The  equations  giving  the  values  of 
5c,  y,  z  are  the  parametric  equations  of  the  same  curves  and  sphere,  respectively. 


10 


PLANE   TRIGONOMETRY 


[§7 


30.  Show  x^  -\-  y^  -\-  z^  —  r^, 

when  X  =  r(cos  d  •  cos  (ft  +  sin  6  •  sin  (f)  •  cos  i/r), 
y  =  r(cos  ^  •  sin  (f)  •  cos  i/r  —  sin  6  •  cos  <f>^, 
z  =  r  ■  sin  <^  •  sin  ■\/r. 

31.  Show  2:^  +  3/^  =  a^,  when  x  =  a  ■  cos^  (f),  y  =  a  •  sin'  <^. 

EXERCISES 

Simplify  the  following  expressions,  construct  the  angle  </>  in 
each  case,  and  read  the  values  of  the  remaining  functions. 

Note.     When  square  roots  are  to  be  taken,  use 
only  the  +  sign. 

1,    sin  (f>  •  sec  ^+  1  =  5. 

Tn  this  exercise,  we  have 

sin  <f>  •  sec  (f>  =  i, 

sin  <b      .        ,       , 

or,  -*-  =  tan  <A  =  4. 

cos  <f> 


/ 

/ 

/'^ 

/ 

% 

/  ^ 

[ 

r 

^^ 

' 

\ 

1 

(c 

) 

Hence, 


■    A        4  .1 

sm  <b  =  — 3:^ ,  cos  ©  =  — :zz » 

Vl7  Vl7 


tan  ^  =  4,  cot  <^  =  t  >  sec  ^  =  VlT,  esc  (f>  = • 

In  the  following  the  values  of  a  single  function  should  be 
found,  then  the  angle  may  be  constructed. 


2.  tan  (^  =  2  —  sin  <f>  •  sec  (f>. 

3.  tan  <f) '  cos  ^  =  |. 

4.  sec  <^  •  cot  ^  =  f . 

5.  sec  <^  =  4  cos  ^. 

6.'  cot  ^  =  3  tan  ^. 

7.  tan  (f>  =  4l  cot  <^. 

8.  cot  ^  =  f  CSC  ^. 

9.  COS  (j>  -r-  sin  ^  =  1^. 

10.  cot  <f>  •  COS  ^  •  tan  ^  =  -  • 

11.  sin^  ^  •  sec  ^  •  cos  <f>  •  csc  ^  =  ^. 

12.  1  — cos2  0=|. 


^Hs.  tan  ^  =  1. 

Ans.  sin  </>  =  f  • 

Ans.  sin  <^  =  4 

^ws.  sec  (f)=  2. 

^ns.  tan  <f)  =  — , 
^       3 

Ans.  tan  <^  =  2. 

J.WS.  cos  (f>  =  ^. 

Ans.  cot  0  =  f . 

^ws.  cos  ^  =  -  • 

^ws.  sin  <f)  =  ^5^. 

^ws.  sin  0  =  f . 


§7]      TRIGOXOMETRIC   FUNCTIONS  OF   ACUTE   ANGLES       11 

13.  sec^  ^  —  1  =  -^^j.  Ans.  tan  <^  =  ■^^. 

14.  2  — 3  sin  ^=  J.  Ans.  sin</)  =  |. 

15.  (1  +  sin^)(l  —  sin^)=  j^g.  Ans.  cos^  =  |. 

16.  tan^  <^  +  1  =  2  sec  <^.  Ans.  sec  (/>  =  2. 

17.  tan^=  2— 2sec<^-^  csc^.  Ans.  tan<^  =  |. 

18.  tan  </)  —  cos^  <^  =  sin^  <^  +  5.  Ans.  tan  <^  =  6. 

19.  sin  <^  =  2*g^  cos  <^  •  sec  0  •  esc  ^.  ^rj«.  sin  ^  =  f  • 

5  tan  <h  X  cos  d)  x  esc  d)      -<  ^  •     ^      -i 

20.    ^- ^- ^  =  1.  ^W8.    sin  9  =  -  • 

4  cot  (^  X  sec  </)  5 

21.  When  tan  «  =  4,  find  value  of 

cos^a 

f\  POS     fi.  "\  •  Sin     CL 

22.  When  sin  «  =  -,  find  value  of — • 

6  tan  a 

1,7-,  1    J2    J       1         £  sin  a  •  tan  ce 

23.  When  cos  a  =  -,  find  value  oi —  • 

2  sec^  a  +  1 

tan^  a 


24.    When  sec  a  =  5,  find  value  of 


cos  a  •  CSC  a 


^  25.    When  cot  a  =  -,  find  value  of  — • 

/  8  csc^  a  —  1 

o  mn    OL    I    f*os  re 

26.  If  sin  rt  =  — ,  what  value  has  ■ ? 

18  l  +  tan2a 

27.  If  tan  «  =  I V8,  what  value  has  3  sin  a  —  4  sin^  a? 

28.  sin  80°  =  1,  find  the  value  of  4  cos^  30°  -  3  cos  30°. 

29.  sec  45°  =  V2,  find  the  value  of 

l-tan2  45° 

30.  sin  30°  =  I,  cos  45°  =  J  V2,  find  the  value  of 

tan  46°  x  cot  30°  h-  (sec  30°  x  sin  45°). 

Change  the  following  functions  so  that  only  sin  a  appears : 

9  cot  a       ,  tan  a 

31.  sm  «  COS"'  a -, 1 — 

cos  a  sin  a      cos  a 

sin  a 


32.  CSC  a  sm  a  +  cos  a  tan  a  — 

cos^a 

33.  tan^  a  -f-  1  —  csc^  a  +  2  sin  a  cos  a. 

34.  sec*  a  —  tan^  a  +  cos^  a  —  cot^  a. 


12  PLANE   TRIGONOMETRY  [§  7 

Change  the  following  so  that  only  tan  a  appears : 

35.  cot  a  +  sec  a  —  esc  a. 

36.  (1  +  cos  a)(l  —  cos  a)  +  tan  a(cot2  a  —  1). 

37.  sin  a  +  cos  a.  +  cos  a  tan  a  +  cot  a  sin  a. 

38.  (1  —  sin  a  cos  a)  h-  (1  +  sin  a  cos  a)  —  cot  a  sin  a. 

Change  the  following  so  that  only  cos  a  appears : 

sin^a  ,  tan «  ^„  1      ,  tana 

39. 1 40.    sec  a -\ — ^ 

cos  a       cot  a  cos  a       sin  a 

41.  cot^  a  +  tan^  «  —  sin^  «  —  cos^  a. 

42.  sin  a  cos  a  tan  a  cot  a. 

43.  Express  sin  a;  in  terms  of  each  of  the  other  functions. 


Thus,  (1)  sina:;  =  Vl  —  cos^a:,  from  identity  (A),  §  5. 


(2)  sin  rr  =  tan  a; -j- Vl  4- tan^  a;,  from  (D),  (F)  and  (B). 

(3)  sin2;= —  -,  from  (G)  and  (C). 

Vl  +  cot^  X 


(4)  sina:=\/l 1-  =  J^^^^^'^from  (A)  and  (F). 

''        sec^a;      ^     sec  a; 

(5)  sina;  = ,  from  (G). 

CSC  a; 

44.  Express  cos  x  in  terms  of  the  other  functions. 

45.  Express  tan  x  in  terms  of  the  other  functions. 

46.  Express  each  function  in  terms  of  sin  x. 

[Note.     Construct  the  angle  x  with  hypotenuse  unity,  the  opposite  side 
sin  X,  and  the  base  Vl  —  sin^  x.] 

47.  Express  each  function  in  terms  of  (1)  cos  a;,  (2)  tan  x^ 

(3)  cot  a;. 

48.  Prove   that   sec  JL(sin  ^  —  cos  J.)  + cscyl(sin^  +  cos^) 

=  sec -4  CSC -4. 

49.  Show  sin  a  +  cos  a  >  1,  0  ^  a  ^  90°.     , 

50.  Show  tan  a  +  cot  a  ^  2. 

51.  Show  sin  a  +  cos  a  ^  V2.  52.    Show  sin  a  cos  n^^. 


Fig.  6. 


§  8]     TRIGONOMETRIC   FUNCTIONS   OF   ACUTE   ANGLES       13 

8.  Functions  of  Particular  Angles.  The  trigonometric  func- 
tions of  certain  angles  may  be  found  from  geometrical  draw- 
ings. 

(1)  Functions  of  45°. 
From  the  drawing,  Fig.  6,  we  have 

sin  45°  =  I V2,  CSC  45°  =  V2, 
cos  45°  =  1 V2,  sec  45°  =  V2, 
tan  45°  =  1,        cot  45°  =  1. 

(2)  Functions  of  30°  and  60°.     From  Fig.  7, 

sin  30°  =  \  =  cos  60°, 

cos  30°  =  ^  =  sin  60°, 

tan  30°  =  ^  =  cot  60°, 
o 

cot  30°  =  VS  =  tan  60°, 

sec  30°  =  f  V3  =  CSC  60°, 

CSC  30°  =  2  =  sec  60°. 

(3)  Functions  of  0°  and  90°.  The  trigonometric  functions 
of  0°  and  90°  are  limiting  values  which  may  be  seen  from  the 
drawing.  Fig.  8. 

As  the  side  y  approaches  0,  the  opposite  angle  approaches  0 
and  the  base  x  approaches  r. 


Pis.  8. 


sinO°=J™o-  =  ^  =  cos90°, 
cosO°=  ^''"  -  =  1  =  sin  90°, 

x  =  r  y  ' 

lim  y 


COtO 


sec 


o lim  ^ 

y=^y 

lim^ 

'  X 


=  00  =  tan  90°, 


0°  =  !""-  =  !  =  CSC  90°, 

x-r  ^, 


lim  ^ 


tan  0°  =  j!L"  -  =  0  =  cot  90°,  esc  0°  =  '™  -  =  oo  =  sec  90° 

y    ^  X  u 


14 


PLANE   TRIGONOMETRY 


[§9 


The  angles  0°,  30°,  45°,  60°,  90°,  occur  so  frequently  that  it 
will  be  found  convenient  to  keep  in  mind  the  numerical  values 
of  the  trigonometric  functions  of  each. 

Tabulate  the  values  of  these  functions  as  shown  in  the  follow- 
ing table : 


^n/ 


e^ 

0° 

80^ 

45^              6(r              00'-"' 

sin  6 

d 

,5- 

.707 

-:  *  * 

/ 

cos  6 

1 

.V6^ 

.707 

,s 

c 

tan  e 

o 

,r7f 

/ 

f*m 

oo 

cot^ 

d=> 

A732 

/ 

.sni 

0 

sec  6 

1 

fJ^^ 

/^ii^ 

2l 

CO 

CSC  6 

- 

f. 

':HH 

/1  /j-o^ 

EXERCISES 

Find  the  values  of  the  following  by  inserting  numerical  values 
of  the  trigonometric  functions  and  reducing  to  simplest  form. 

1.  5  cos  60°  -  3  sin  30°  +  tan  45°.  Am.  2. 

2.  5  cos  30°  +  4  cos  60°  -  5  sin  60°.  Ans.  2. 

3.  8  tan  30° -4  cot  45°  + cos  90° -8  cot  60°.  Ans.  -4. 

4.  10  cos  30°  +  16  cos  60°  -  5  V3.  An».  8. 

5.  (4  tan  45° -11  tan  60°) (4  cot  45° +  11  cot  30°).  Ans.  -347. 

6.  (16  tan  0°  +  10  cot  90°  +  6  sin  30°)  cos  90°.  Ans.  0. 

7.  sin  30°  X  cos  0°  x  tan  45°  x  sec  60°. 

8.  {x  +  y)  cos  0°  -  (2;  -  «/)  tan  45°  -  2  3/  sin  30°. 

9.  (a;  +  yy  sin  60°  -{x-  yY  cos  30°  -  2  xy  tan  60' 

10.  (a  +  6)  sec  60°  +  (a  -  5)cos  90°  +  a  tan  90°. 

11.  «  sin  30° -(a -J)  tan  45° -6  sin  30°. 

12.  (a  sin  60° -5  cos  30°)  X  CSC  60°. 


Ans.  1. 

Ans.  y. 

Ans.  0. 

Ans.  30. 
Ans.  1(5  —  a). 
A71S.  a  —  h. 


9.  Tables.  For  convenience  in  the  numerical  calculations 
which  follow  on  p.  26  a  condensed  table  of  trigonometric 
functions,  true  to  three  decimals,  will  now  be  inserted.  By 
use  of  this  table  approximate  results  may  be  obtained  for  nu- 
merical problems. 


OL 


§  9]      TRIGONOMETRIC    FUNCTIONS   OF   ACUTE   ANGLES       15 
NATURAL   TRIGONOMETRIC   FUNCTIONS 


A  SOLE 

i        Sis 

Csc 

T.4N 

Cot. 

Sec 

Cos 

0° 

i    0.000 

00 

0.000 

00 

1.000 

1.000 

90° 

1 

0.017 

57.30 

0.017 

57.29 

1.000 

1.000 

89 

2 

i    0.035 

28.65 

0.035 

28.64 

1.001 

0.999 

88 

3 

0.052 

19.11 

0.052 

19.08 

1.001 

0.999 

87 

4 

0.070 

14.34 

0.070 

14.30 

1.002 

0.998 

86 

5^ 

0.087 

11.47 

0.087 

11.43 

1.004 

0.996 

85° 

6 

0.105 

9.567 

0.105 

9.514 

1.006 

0.995 

84 

7 

0.122 

8.206 

0.123 

8.144 

1.008 

0.993 

83 

8 

0.139 

7.185 

0.141 

7.116 

1.010 

0.990 

82 

9 

0.156 

6.392 

0.158 

6.314 

1.012 

0.988 

81 

10^ 

0.174 

5.759 

0.176 

5.671 

1.015 

0.985 

80° 

11 

0.191 

5.241 

0.194 

5.145 

1.019 

0.982 

79 

12 

0.208 

4.810 

0.213 

4.705 

1.022 

0.978 

78 

13 

0.225 

4.445 

0.231 

4.331 

1.026 

0.974 

77 

14 

0.242 

4.134 

0.249 

4.011 

1.031 

0.970 

76 

15^ 

0.259 

3.864 

0.268 

3.732 

1.035 

0.966 

75° 

16 

0.276 

3.628 

0.287 

3.487 

1.040 

0.961 

74 

17 

0.292 

3.420 

0.306 

3.271 

1.046 

0.956 

73 

18 

0.309 

3.236 

0.325 

3.078 

1.051 

0.951 

72 

19 

0.326 

3.072 

0.344 

2.904 

1.058 

0.946 

71 

20"^ 

0.342 

2.924 

0.364 

2.747 

1.064 

0.940 

70° 

21 

0.358 

2.790 

0.384 

2.605 

1.071 

0.934 

69 

22 

0.375 

2.669 

0.404 

2.475 

1.079 

0.927 

68 

23 

0.391 

2.559 

0.424 

2.356 

1.086 

0.921 

67 

24 

0.407 

2.459 

0.445 

2.246 

1.095 

0.914 

66 

25° 

0.423 

2.360 

0.466 

2.145 

1.103 

0.906 

65" 

26 

0.438 

2.281 

0.488 

2.050 

1.113 

0.899 

64 

27 

0.454 

2.203 

0.510 

1.963 

1.122 

0.891 

63 

28 

0.469 

2.130 

0.532 

1.881 

1.133 

0.883 

62 

29 

0.485 

2.063 

0.554 

1.804 

1.143 

0.875 

61 

30= 

0.500- 

2.000 

0.577 

1.732 

1.155 

0.866 

60° 

31 

0.515 

1.942 

0.601 

1.664 

1.167 

0.857 

59 

32 

0.530 

1.887 

0.625 

1.600 

1.179 

0.848 

58 

33 

0.545 

1.836 

0.649 

1.540 

1.192 

0.839 

67 

34 

0.559 

1.788 

0.675 

1.483 

1.206 

0.829 

56 

35° 

0.574 

1.743 

0.700 

1.428 

1.221 

0.819 

56° 

36 

0.588 

1.701 

0.727 

1.376 

1.236 

0.809 

54 

37 

0.002 

1.662 

0.754 

1.327 

1.252 

0.799 

53 

38 

0.616 

1.624 

0.781 

1.280 

1.269 

0.788 

52 

39 

0.629 

1.589 

0.810 

1.235 

1.287 

0.777 

51 

40^ 

0.643 

1.556 

0.839 

1.192 

1.305 

0.766 

50° 

41 

0.656 

1.524 

0.869 

1.150 

1.325 

0.756 

49 

42 

0.669 

1.494 

0.900 

1.111 

1.346 

0.743 

48 

43 

0.682 

1.466 

0.933 

1.072 

1.367 

0.731 

47 

44 

0.695 

1.440 

0.966 

1.036 

1.390 

0.719 

46 

45° 

1    0.707 

L414 

1.000 

1.000 

1.414 

0.707 

46° 

Cos 

SF.r, 

Cot 

Tan- 

Cso 

Sin 

Angle 

CHAPTER   II 


SOLUTION  OF  RIGHT  TRIANGLES 

A  right  triangle  is  known  when  a  side  and  any  other  part 
are  known.  The  use  of  trigonometric  functions  enables  us  to 
compute  the  unknown  parts. 

We  have,  Fig.  9,  the  following 
B     relations  : 

(1)  a^  +  h^  =  c^-, 

(2)  sin  ^  =  ^  =  cos  B, 

^  c 

(3)  cos  ^  =  -  =  sin  JS, 


10.    Fundamental  Formulas. 


(4)  tan^=^  =  cotJS, 

The  first  of  these 
formulas  is  a  statement  of  the  Pythagorean  Theorem  ;  (2),  (3), 
(4)  are  results  from  the  definition  of  sine,  cosine,  tangent. 

With  any  two  dimensions  given  (not  A  and  B^  the  other 
three     dimensions     may    be 
found. 

Examples.   1.     Given  ^  =  37°, 
c  =  12  in. ;  find  a  and  b. 

Solution.     From  formula  (2), 

a  =  c  sin  ^  =  12  X  sin  37°. 
From    the    approximate    table, 
p.  15, 

sin  37°  =  .602. 
Therefore, 

o  =  12  X  .602  =  7.2  in. 
From  (3), 

b  =  ccos  37°  =  12  X  .799 

=  9.59  in.  (a) 

16 


. 

,^ 

/ 

/ 

/ 

-^^> 

y 

c 

i> 

1 

/ 

y 

,/ 

y] 

7° 

^ 

? 

A 

b 

( 

a 

) 

§11] 


SOLUTION   OF   RIGHT   TRIANGLES 


17 


2.    Given  a  =  20  ft.,  c  =  35  ft.,  find  angle  A. 
20 


Solution,     sin  A 


3.3 


,571. 


From  table,  p.  15,  the  angle  whose  sine  is  .571  is  34°  50'  (approximately). 
Hence  A  =  34°  50'. 

11.  Projections.  Let  perpendiculars  fall  from  two  points, 
A  and  B  (Fig.  10)  upon  any  line  Z,  intercepting  PQ.  PQ  in 
called  the  projection  of  the  line  AB  upon  L.  This  manner  of 
projection  is  called  orthogonal. 


(1)  JIori.on,al  an,  ..ticoX /^^n.  Any  line  AB  n,a, 
be  projected  orthogonally  in  any  direction  ;  the  usual  projec- 
tions, however,  are  upon  the  horizontal  PQ^  or  upon  the  verti- 
cal RS. 

(2}  Laivs  of  orthogonal  'projection.      The  orthogonal  projec- 
tion of  any  line  upon  another  line  involves  the  base  and  alti- 
tude   of    a    right  triangle,  „ 
with  the  hypotenuse  given. 
Hence, 

(a)   The  horizontal  pro- 
jection, 

a;  =  r  .  cos  9, 

(5)  The  vertical  projec- 
tion, A 
?/  =  »'.  sill  9. 

Theohem.  The  horizontal  projection  of  any  line  segment 
equals  the  length  of  the  segment  multiplied  hy  the  cosine  of  the 


Fig.  11. 


18 


PLANE   TRIGONOMETRY 


[§§  12-13 


angle  of  inclination;  the  vertical  projection  equals  the  length  of 
the  segment  multiplied  hy  the  sine  of  the  angle  of  inclination. 

12.  Components.  Forces,  velocities,  and  accelerations  may  be 
represented  by  directed  lines.  A  line  of  given  length  and  fixed 
direction  is  sometimes  called  a  vector.  Thus,  in  Fig.  12,  T'^is  a 
vector,  also  F^,  Vy. 


Y 

J 

I 

B 

"y 

^r       \a 

Vy 

P 

^                       v^ 

^C 

0 

X 

Fig.  12. 

Suppose  a  body  is  moving  with  a  velocity  V  in  the  path  AB, 
Fig.  12,  making  an  angle  a  with  the  horizontal  OX;  we  may 
require  the  velocity  in  the  horizontal  A  0  (the  horizontal  com- 
ponent of  the  velocity,  F^),  also  the  vertical  velocity  CB  (the 
vertical  component  of  the  velocity,  Vy).  These  components  are 
projections  of  AB  upon  the  horizontal  and  vertical,  respectively. 

AC  =  AB cn^  a,  or  Vx=  Fcosa; 
CB  =  AB  sin  a,  or  Vy  =  Fsin  a. 

Squaring  and  adding, 

.     F«2  +  Fj,2  =  f2(cos2  a  +  sm2  a)  =  F^. 

Theorem:.  The  sum  of  the  squares  of  the  horizontal  and  ver- 
tical components  of  a  velocity  equals  the  square  of  the  velocity. 

13.  Resultant.  If  two  forces  act  upon  a  particle  at  A,  the 
one,  AC,  acting  horizontally,  the  other,  AB,  acting  vertically, 
the  particle  at  A  will  be  moved  along  the  diagonal  of  the 
parallelogram  to   B,  Fig.   13.     The  line  AB  is  called  the  re- 


§13] 


SOLUTION   OF   RIGHT   TRIANGLES 


19 


sultant  of  the  forces  represented  by  the  two  lines  A  O  and  AD. 
The  forces  A  C  and  AD  need  not  act  at  right  angles,  but  what- 
ever their  angle,  the  resultant  will  be  the  diagonal  of  the  par- 
allelogram constructed  upon  AC  and  AD. 


Fig.  13. 

The  relations  between  resultant  and  components  which  act 
at  right  angles  to  each  other  involve  the  simple  trigonometric 
properties  of  right  triangles. 

Examples,  l.  A  train  of  cars  is  moving  northeast  with  a 
velocity  V  of  40  mi.  per  hour.  Find  its  rates  of  travel  east 
and  north. 

Solution.  Let  the  vector  V  represent  the  given  velocity.  Then  V^,  Vy 
are  the  velocities  east  and  north,  respectively.     See  Fig.  12. 

V^  =  V  cos  45°  =  40  X  I  V2  =  20\/2, 

Fy  =  F  sin  4.7'  =  40  x  |  V2  =  20  V2. 

Hence,  the  velocities  east  and  north  are  equal,  20  V2  mi.  per  hour. 

2.  A*  point  is  moving  with  a  velocity  of  25  ft.  per  second 
along  a  line  making  an  angle  of  38°  with  the  horizontal.  Find 
the  horizontal  component.  Ans.  19.7  ft. 

3.  Find  the  velocity  of  a  point  moving  at  an  angle  of  64° 
with  the  horizontal,  if  the  vertical  component  of  the  velocity  be 
100  ft.  per  second.  Avis.  111.3  ft. 

4.  A  point  describes  a  circle  of  radius  20  inches  at  a  uniform 
rate  of  two  revolutions  per  minute.  Find  the  distance  from 
the  centre  of  the  circle  to  the  projection  of  the  point  upon  a 


20 


PLANE  TRIGONOMETRY 


[§13 


horizontal  diameter  5  seconds  after  passing  the  extremity  of 
that  diameter.  Ans.  10  in. 

5.  A  force  F  of  100  lb.  is  applied  to  a  block  at  the  point  0, 
see  Fig.  (6).  If  the  force  makes  an  angle  of  60°  with  the  hori- 
zontal, what  force  tends  to  draw  the  block  horizontally  (OC)  ? 
What  force  tends  to  lift  the  block  upwards  (  Oi>)  ? 

Solution.  Here  the  solutions  called  for 
are  the  horizontal  component  F^  and  the 
vertical  component  Fy  of  the  force  F. 

F^  =  Fx  cos  a  =  100  cos  60°  =  50  lb., 
Fy^  F  X  sin  «  =  100  sin  60"  =  86.6  lb. 

6.  A  balloon  rising  vertically 
at  a  uniform  velocity  of  704  ft.  per 
minute  encounters  a  wind  blowing 
horizontally  at  the  rate  of  24  mi. 
per  hour.  Find  the  angle  at  which 
the  balloon  will  rise  and  its  velocity 
after  meeting  the  wind. 

Solution.     The  angle  required  is  represented  by  a  in  Fig.  c,  the  velocity 

of  wind  and  vertical  velocity  of  the  balloon  by  OB  and  OA,  respectively. 

.  704      1 

tan  a  — =-, 

2112     3 

a  =  IS''  26'. 

V  =  2112  X  sec  « 

=  2112  X  1.054  =  2226  ft. 


-4-4 

-4^^+- 

-+^-4— 

::::±: 

# 

t"" 

-  I'll 

-  - — —y 

i,^^ — 

-n- 

ll 

i        iy 

i  ■  ! 

-~CbXi 

^-+f 

--t4-T 

-—f— 

-•<7^ 

,-i^r 

-t-i-L- 

1 ;  ■ 

._t-^ 

^^"iiii 

'  ■  1 

'  :  ■ 

. ;  i  M  i 

i  ;  :  1 

--n- 

-y^ 

Va"-=-60— 

==0- 

'  c  ;  > 

.  .     1 

— - 

1    -:p.-| 

r-ar    ; 

1 . .  ij 

IfH 

1 1 1  Hill  1 

W^t- 

Si 

- .,,    .- 

:-:::"::Cl ::::"  :::::2li5Pl;::::::::::::EI::"  -::.: 

7.  A  force  of  400 
lb.  acting  in  a  direc- 
tion inclined  40°  from 
the  horizontal  is  ap- 
plied to  a  heavy  block.  Find  the  force  which  tends  to  move 
the  block  (1)  horizontally,  (2)  the  force  which  tends  to  lift  the 
block  vertically.  Ans.   (1)  306.4  lb.,  (2)  257.2  lb. 

8.  The  horizontal  and  vertical  components  of  a  force  acting 
upon  a  heavy  block  are  160  lb.  and  120  lb.  respectively. 
Find  the  force  and  its  direction  of  action. 

Ans.  200  lb.,  36°  52'. 


§§  14,  15]  SOLUTION   OF   RIGHT   TRIANGLES  21 

14.   Projected  Areas.     Any  plane  area  ABCD  inclined  to  a 

C 


Fig.  14. 

plane  LM  at  an  angle  6  may  be  projected  orthogonally  upon 
LM  iuio  the  area  ABFE. 

ABFE  =  ABCD  x  cos  9 

Laiv  of  projection.  The  projection  of  a  given  plane  area  upon  a 
plane  equals  the  given  area  multiplied  hy  the  cosine  of  the  angle 
of  inclination  of  the  two  planes. 

Examples,  i.  How  much  horizontal  area  will  be  covered 
by  60  sq.  yd.  of  roofing,  the  roofing  making  an  angle  of  45° 
with  the  horizontal  ?  Ans.   60  cos  45°  =  42.42  sq.  yd. 

2.  Find  the  roofing  required  to  cover  a  horizontal  space 
15  ft.  by  24  ft.,  the  roofing  rising  at  an  angle  of  40°. 

Ans.  15  X  24  X  sec  40°  =  469.8  sq.  ft. 

15.  Area  of  the  Right  Triangle.  The  area  of  a  right  triangle 
may  be  expressed  in  various  ways.  Let  ^=area;  then  we 
have, 

(1)  ir  =  5«x6=^-5ac  cos  A  =  -^hc  sin  A. 

(2)  K  :=  \  Ir-  tan  .1  =  |  a^  ^^^  ^ 

(3)  ^  =  I  c^  sin  ^  cos  A  =  \  c^  sin  B  cos  B. 


22 


PLANE   TRIGONOMETRY 


[§16 


In     (1)    a  cos  A  =  b  sin  A  =  p  ;    and    in    (2)    b  tan  A  =  a, 
a  tan  B=b. 


Examples.     Find  the  area  of  each  of  the  following  right 
triangles,  the  lettering  being  shown  in  Fig.  15 : 

1.    «  =  20,  J.  =  35°.  6.    c  =  75,   a  =  2b. 


2.  (?  =  65,   J5  =  28°. 

3.  a  =  100,   ^  =  30°. 

4.  6=42,    5=67°. 

5.  (?  =  35,   A  =  3  5. 


7.  JO  =27,   ^  =  48°. 

8.  j»  =  45,   B=15°. 

9.  6  =  48,    a  =  52. 
10.  a  =  65,  5e=13J. 


/y~ 

\ 

at 

h 

\a 

/      ^P' 

a/    \ 

6 
Fig.  16. 


16.    The  Isosceles  Triangle. 

Since  the  isosceles  triangle 
may  be  divided  into  two  equal 
right  triangles  by  a  perpen- 
dicular from  the  vertical  angle 
B  (see  Fig.  16)  upon  the  base, 
the  solution  depends  only  upon 
the  solution  of  a  right  tri- 
angle. 

Examples,  l.  Find  the 
base  b  and  area  of  an  isosceles 
triangle  when  the  vertical 
angle  B  is  80°,  and  each  of 
the  equal  sides  is  36  ft. 


§§  16-17] 


SOLUTION   OF   RIGHT   TRIANGLES 


23 


Solution.  Construct  an 
approximate  triangle,  Fig.  d. 

(1)  b  is  the  projection  of 
the  equal  sides  on  the  hori- 
zontal.    Heuce, 

h  =  2a  X  sin  ^  =  2  x  36  x 

sin  40°  =  2  X  36  X  .643  = 
46.3  ft. 

(2)  Area  =  |  6  •  h  =  ^  abx 
cos  40°  =  638.2  sq.  ft. 

2.    Find  the  length  of 

rafters,  height  of  ridge- 


pole, and  area  of  gable  from  the  data  shown  in  the  drawing. 


(1)  Length  of  rafter  =  16  x  sec  37°  =  16  x  1.252  =  20.03  ft. 

(2)  Height  A  =  16  x  tan  37"  =  16  x  .754  =  12  ft. 

(3)  Area  of  gable  =  16  x  /«  =  192  sq.  ft. 

17.  Notations  of  Direction.  Tf  Off,  Fig  17,  be  a  horizontal, 
the  angle  POff=(f)  is  the  angle  of  elevation  of  P :  the  angle 
QOH=y^  is  called  the  angle  of 
depression  of  Q.  A  depression 
angle  could  also  be  considered 
as  a  negative  elevation  angle. 

The  direction  of  a  line  OP  with 
respect  to  a  north  and  south  line 
NS,  Fig.  18,  is  read  North  a  de- 
grees East,  N.  a°  E.  ;  the  same  Fig.  17. 
direction  with  regard  to  the  east 
and  west  line  EWcowXdi  be  read  East  /9  degrees  North,  E.  ^°  N. 


24 


PLANE   TRIGONOMETRY 


[§17 


Similar  readings  for  directions  may  be  applied  to  south  and 


W-^ 


west  lines.     A  direction  of  a  line  given  as  above  is  called  the 
hearing  of  the  line  when  read  from  N.  or  S. 

Examples,    i.    If  the  angle  of  elevation  of  the  sun  be  35°, 

how  high  is  a  pole 
whose  shadow  upon 
the  ground  is  40  ft.  ? 

Solution.  (1)  Construct 
drawing  showing  data. 

(2)  Select  formula  (4), 
p.  16, 

A 
40' 

A  =  40  X  tan  35° 
=  40  X  0.7 
=  28  ft. 


tan  35°  = 


2.  Find  the  radius  of  a  circle  in  which  a  chord  30  in. 
long  subtends  a  20°  arc.  x41so  find  the  area  of  the  triangle 
formed  by  the  chord  and  the  radii  to  its  extremities. 

Solution.  (1)  Construct  figure ;  draw  radius  perpendicular  to  middle  of 
chord.     Let  B  be  the  middle  point  of  the  chord. 


Angle  AOB  =  10°,  AB  =  15  in. 


(2) 


sin  A  OB 


AB 
R  ' 


R  =  AB  ---  sin  AOB, 


or, 


R  =  ABx  CSC  AOB. 


§§  17-18] 


SOLUTION   OF   RIGHT   TRIANGLES 


25 


(3)  Substitute  the  values  AB  =  15,  AOB  =  10°,  and  we  have 

R  =  ld  X  CSC  lU"  =  15  X  5.759  =  86.38  in. 


(4)  The  area  is  given  by 

K  =  I  base  x  altitude.     05  =  15  x  cot  10°  =  85.06. 
Then  K  =  15  x  OB  =  1275.9  sq.  in. 

18.  To  solve  Right  Triangles.  In  the  solution  of  right  tri- 
angles the  beginner  should  provide  himself  with  (1)  a  gradu- 
ated ruler,  (2)  squared  paper,  and  (3)  a  protractor  graduated 
to  degrees.  With  this  equipment  a  triangle  approximating 
very  closely  to  any  given  data  may  be  constructed.  An 
approximate  construction  will  enable  the  student  to  detect  any 
considerable  error  in  calculation. 

In  solving  a  right  triangle  the  following  directions  are  sug- 
gested :  (1)  Draw,  on  squared  paper,  a  figure  as  accurately/  as 
possible  from  the  given  data,  and  estimate  approximate  values  for 
the  unknown  parts.  (2)  Select  formulas,  §  10,  each  of  which 
contains  one  unknoivn.  (3)  Substitute  the  given  data  in  the 
proper  formula,  using  approximate  values  of  the  trigonometric 
functions,  p.  15,  and  solve  for  the  unknoivns.  (4)  Check  results 
by  using  some  formula  7iot  employed  in  the  calculation. 

In  many  ordinary  measurements  approximate  results  only 
are  desired.  The  following  list  of  exercises  involving  the 
right  triangle  is  inserted  to  give  practice  in  approximating 
results.  The  trigonometric  functions  found  on  p.  16  give 
approximations  to  three  decimals,  and  when  employed  in  solu- 
tion will  give  results  true  to  one  or  two  decimals.  When 
greater  accuracy  is  desired,  the  more  complete  tables  of  the 
natural  functions  should  be  used.  For  still  more  accurate 
results,  the  logarithmic  tables,  explained  in  Chapter  VI,  should 
be  employed. 


26 


PLANE   TllIGONOMETRr 


[§18 


EXERCISES 


Solve  the  following  right  triangles,  giving  results  to  nearest 
tenth  and  nearest  minute.  Use  Approximate  Tables  of  trigo- 
nometric functions,  p.  15,  for  solutions  of  Ex.  1  to  20,  approxi- 
mating angles  to  minutes.     For  areas  employ  formulas  of  §  15. 


No. 
1 

Data 

Answers 

Area  =  A' 

a  =13 

6  =40 

^=18° 

5  =  72° 

c=42 

260 

2 

a  =20 

b  =8 

A  =  68°  12' 

5  =  21°  48' 

c  =21.5 

80 

-/ 

3 

a  =80 

6  =30 

A  =  69°  27' 

B  =  20°  33' 

c  =  85.4 

1200 

4 

a  =16.16 

b  =25.28 

A  =  32°  35' 

B  =  57°  25' 

c  =  30 

204.3 

5 

«  =10 

c   =50 

^  =  11°  32' 

B  =  78°  28' 

6  =49 

245 

6 

a  =71 

c  =78 

A  =  65°  32' 

7?  =  24°  28' 

ft  =  32.3 

1146.6 

7 

a  =84.9 

c   =93.5 

A  =  65°  14' 

B  =  24°  46' 

ft  =39.17 

1662.8 

8 

c   =42 

A  =  81°  30' 

5  =  8°  30' 

a  =41.5 

ft  =6.2 

128.9 

9 

c   =100 

A  =  36° 

i?  =  54° 

a  =58.8 

ft  =  80.9 

2378.5 

10 

c   =67.7 

A  =  23°  30' 

B  =  66°  30' 

a  =27 

ft  =  62.08 

838.1 

11 

c   =250 

5  =  47° 

A  =  43° 

ffl  =  170.5 

6  =  182.8 

15583.7 

12 

c   =400 

b   =240 

A  =  53°  8' 

B  =  36°  52' 

a  =  320 

38400 

'V* 

13 

iT  =55.42 

^  =  30° 

a  =8 

ft  =8\/3 

c  =16 

14 

^=28.93 

A  =  26°  45' 

a  =5.4 

6  =10.7 

c  =12 

15 

K  =  145.8 

6  =18 

^=42° 

c   =24.2 

a  =  16.2 

16.  A  monument  283  ft.  high  casts  a  shadow  100  ft.  long 
upon  the  ground.  Find  the  angle  of  elevation  of  the  sun  at 
that  instant.  Ans.    70°  32'. 

'J  17.  A  ladder  30  ft.  long  resting  upon  the  ground  reaches  a 
point  24  ft.  high  upon  a  vertical  wall.  Find  the  angle  of  ele- 
vation of  the  ladder.  Ans.  53°  6'. 

18.  Gable  rafters  20  ft.  long  project  2  ft.  beyond  the  walls 
of  a  house  and  are  set  with  a  pitch  (angle  of  elevation)  of  40°. 
Find  the  height  h  of  the  ridge-pole  and  the  width  of  the  house. 

Ans.  A  =  11.57  ft.,  w=  27.57  ft. 

19.  Find  the  bearing  of  a  road  which  leads  to  a  point  5  mi. 
east  and  8  mi.  north.  Ans.  N.  32°  E. 

20.  A  rectangle  is  98  by  148.  Find  the  angle  made  by  a 
diagonal  with  the  longer  side.  Ans.   33°  30'. 

21.  The  sides  of  an  isosceles  triangle  are  30,  45,  45,  respec- 
tively.    Find  the  angles.     [Note.    Use  four-place  tables.] 

Ans.  70°  32',  38°  56'. 


§  18]  SOLUTION   OF   RIGHT   TRIANGLES  27 

22.  Find  the  radius  of  a  circle  in  which  a  100  ft.  chord  sub- 
tends an  angle  of  18°  at  the  centre.  Ans.   319.69  ft. 

23.  A  chord  50  in.  long  subtends  an  angle  of  36°  at  the 
centre.  Find  the  radius  M  of  the  circle  and  the  area  A  of  the 
inscribed  square.  Ans.  R=  80.9  in.,  A=  13,089.6  sq.  in. 

1.      24.    Find  the  height  of  a  tree  which  casts  a  horizontal  shadow 
60  ft.  long  when  the  sun's  elevation  is  65°.  Ans.  128.67  ft. 

,,V^  25.    Find  the  angle  of  inclination  of  the  faces  of   a  wedge  /^ 
whose  base  is  3  in.  and  whose  face  is  14  in.  long.    Ans.  12°  18'. 

26.  The  exterior  angle  between  two  500  ft.  tangents  is  72°. 
Find  the  radius  of  the  circle.  Ans.  688.2  ft. 


/- 


y^ 


27.  The  length  of  a  kite-string  is  200  m.  Find  the  height 
of  the  kite  when  its  angle  of  elevation  is  34°.       J.ws.111.84  m. 

28.  The  radius  of  a  circle  is  5  cm.  and  the  length  of  a  chord 
is  4  cm.     Find  the  angle  subtended  by  the  chord  at  the  centre. 

Ans.  47°  9'  20". 

29.  Find  the  radius  of  a  circle  inscribed  in  an  equilateral 
triangle  whose  perimeter  is  42  cm.  Ans.  4.04  cm. 

30.  What  is  the  height  of  a  tower  if  a  16  ft.  flagpole  upon 
the  top  of  the  tower  subtends  an  angle  of  4°  at  a  point  on  the 
ground  and  the  angle  of  elevation  of  the  bottom  of  the  pole  is 
40°  ?  Ans.  106  ft. 

31.  At  a  certain  point  the  angle  of  elevation  of  a  mountain  peak 
is  27°  ;  at  a  distance  of  2  mi.  farther  away  in  the  same  direction 
its  elevation  angle  is  25°.  Find  the  horizontal  distance  from 
the  first  point  of  observation  to  the  peak.  Ans.   21.58  mi. 

32.  Two  consecutive  milestones  on  a  level  road  in  the  same 
vertical  plane  as  a  tower  have  depression  angles  of  42°,  3°, 
respectively,  from  the  top  of  the  tower.  Find  the  height  of 
the  top  of  the  tower  and  its  horizontal  distance  from  the  nearer 
milestone. 

Atis.  h  =  293.8  ft.  or  261.4  ft.,  (^=  326.3  ft.  or  290.3  ft. 

(k>33.  a  tower  stands  upon  the  same  plane  as  a  house  whose 
height  is  60  ft.  The  elevation  and  depression  of  the  top  and 
bottom  of  the  tower  from  the  top  of  the  house  are  41°  and  35°, 
respectively.     Find  the  height  of  the  tower.       Ans.  134.49  ft. 


'L. 


.\^^^  -V^-'^Jd^  J...UJUM  ^ 


28  PLANE   TRlGONOMETRr  [§  18 

/    -~7       34.    From  a  point  10  ft.  above  the  water,  the  angle  of  eleva- 

/    tion  of  the  top  of  a  tree  standing  at  the  edge  of  the  water  is  46°, 

while  the  depression  angle  of  its  image  in  the  water  is  50°. 

Find  the  height  of  the  tree,  and  its  horizontal  distance  from  the 

point  of  observation.  Ans.  A  =  142.5  ft.,  d!  =  128ft. 

/  -pt     35.    In  measuring  the  width  of  a  river  a  line  AB  is  measured 
^'^^    240  ft.  along  one  bank.     A  perj)endicular  to  AB  at  A  is  erected 
which  locates  a  point  O  upon  the  opposite  bank,  and  the  angle 
ABC'\&  found  to  be  65°.     Find  the  width  AC  oi  the  stream. 

Ans.  514.68  ft. 

36.  Two  towers  upon  the  same  horizontal  plane  are  of  such 
heights  that  their  elevation  angles  from  a  point  midway  be- 
tween them  are  40°,  60°,  respectively.  Find  the  ratio  of  their 
heights.  Ans.  8391  h- 17,321. 

37.  From  each  of  two  stations  a  mile  apart  upon  a  north  and 
south  road,  the  angle  of  elevation  of  a  balloon  is  observed  to  be 
30°,  and  its  bearings  are,  respectively,  N.E.  and  S.E.  Find 
the  height  of  the  balloon.  Ans.  2155.5  ft. 

38.  A  balloon  is  exactly  over  the  middle  point  between  two 
cities.  The  balloon  is  a  mile  high  and  the  distance  between  the 
cities  subtends  an  angle  of  136°  at  the  balloon.  Find  the  dis- 
tance between  the  cities  and  the  distance  of  the  balloon  from 
either  of  them.  Ans.   4.95  mi.,  2.67  mi. 

39.  Find  the  height  of  a  tree  if  the  angle  of  elevation  of  its 
top  changed  from  36°  to  42°  on  walking  toward  it  75  ft.  in  a 
horizontal  line  through  its  base.  Ans.   282.16  ft. 

40.  A  hill  rises  uniformly  4  ft.  in  a  horizontal  distance  of  85 
ft.  What  is  the  difference  in  elevation  of  two  points  3500  ft. 
apart,  the  distance  being  measured  along  the  ground  ? 

Ans.  164.1  ft. 

41.  What  is  the  angle  of  incline  of  a  railroad  track  if  it  rises 
30  ft.  in  a  horizontal  distance  of  a  mile  ?  Ans.  19'  32". 

42.  What  is  the  bearing  of  a  road  which  leads  to  a  point 
14  mi.  east  and  8  mi.  north  ?  Ans.  N.  60°  15'  20"  E. 

43.  If  the  radius  of  the  earth  (3956  mi.)  subtends  57'  at  the 
moon,  what  is  the  moon's  distance  from  the  earth  ? 

Ans.  238,300  mi. 


§  18]  SOLUTION  OF  RIGHT  TRIANGLES  29 

44.  Find  the  radius  of  one's  horizon  if  he  be  located  1320  ft. 
above  the  earth.  How  large  when  located  2  mi.  above  the 
earth?  Ans.  44.48  mi.,  125.8  mi. 

45.  How  high  above  the  earth  must  one  be  in  order  to  see  a 
point  located  on  the  surface  50  mi.  away?  Ans.   1668.2  ft. 

V  46.    The  radius  of  a  circle  is  240  ft.     Find  the  perimeter  of 
a  regular  inscribed  pentagon.  Ans.   1410.72  ft. 

V  47.    The  radius  of  a  circle  is  85  ft.     What  is  the  area  of  the 
regular  inscribed  decagon?  Ans.   21,234.27  sq.  ft. 

48.  Find  the  perimeter  of  a  regular  dodecagon  inscribed  in  a 
circle  whose  radius  is  25  in.  Ans.  155.25  in. 

"  49.    What  is  the  radius  of  a  circle  inscribed  in  an  equilateral 
triangle  whose  perimeter  is  99  ft.?  Ans.   9.52  ft. 

50.  The  area  of  a  regular  pentagon  inscribed  in  a  circle  is 
475.53  sq.  cm.  Find  the  area  of  a  regular  decagon  inscribed  in 
the  same  circle.  Ans.  587.8  sq.  cm. 

51.  The  area  of  a  regular  pentagon  is  441  sq.  ft.  Find  the 
apothem ;  also  find  the  radius  of  the  circumscribed  circle. 

Ans.  11  ft.;  13.6  ft. 

52.  What  is  the  length  of  a  diagonal  joining  the  first  and 
fourth  vertices  of  a  regular  polygon  of  12  sides  inscribed  in  a 
circle  whose  radius  is  30  ft.  ?  Ans.  42.426  ft. 

53.  If  i?  =  radius  of  a  circle,  show  that  the  area  of  a  regular 

180°        180° 
inscribed  polygon  of  n  sides  h  A  =  nR^  sin cos 


n  n 

54.  From  the  result  in  Ex.  53  construct  a  table  showing,  in 
terms  of  jB,  areas  of  the  regular  inscribed  polygons  of  3,  4,  5, 
9,  10,  12,  sides. 

55.  Show  that  the  area  of  a  regular  circumscribed  polygon 
of  n  sides  is  given  by  A  =  nM^  tan 

56.  A  train  moving  at  a  uniform  speed  of  40  mi.  per  hour 
on  a  straight  track  passes  a  station  A  at  noon ;  at  2:15  o'clock 
p.  M.  it  has  arrived  at  a  station  B,  56  mi.  farther  north.  How 
far  east  of  A  is  B,  and  what  is  the  bearing  of  the  road  ? 

Ans.  70.46  mi.:  N.  51°  31' 30"  E. 


30  PLANE   TRIGONOMETRY  [§18 

57.  A  boat  running  at  a  rate  of  10  mi.  per  hour  starts  to 
steam  directly  across  a  river  one  mile  in  width.  If  the  water 
be  flowing  uniformly  at  a  rate  of  4  mi.  per  hour,  find  the  point 
j^t  which  the  boat  will  land,  and  its  velocity  in  the  water. 

An8.  0.4  mi.  downstream,  10.77  mi.  velocity. 

58.  Find  the  projection  of  the  altitude  of  an  equilateral  tri- 
angle upon  a  side.     Let  a  =  side.  Ans.  |  a. 

59.  A  line  extends  N.  20°  E.  125  rd.  from  a  point  A.  Find 
its  projection  upon  a  line  extending  N.  60°  E.  from  A. 

Ans.  95.75  rd. 

60.  Find  the  projection  of  a  line  450  ft.  long  running  N. 
47°  E.  upon  a  line  running  E.  15°  S.  Ans.  238.45  ft. 

61.  Two  forces  of  160,  120  lb.  act  upon  a  heavy  body,  the 
first  at  an  angle  of  30°  with  the  horizontal,  the  second  at  an 
angle  of  75°.  Find  the  total  forces  which  tend  to  move  the 
body  (1)  horizontally,  (2)  vertically. 

Am.  (1)  169.6  lb.,  (2)  195.9  lb. 

^7^'  62.  Find  the  number  of  square  yards  in  a  conical  tent  with 
circular  base,  the  vertical  angle  being  00°,  and  the  centre  pole 
12  ft.  high.      [Take  7r  =  ^2..-]  ^^g.   33.5  gq.  yd. 

63.  Find  the  area  in  acres  of  the  following  tract  of  land: 
starting  from  a  point  A^  the  boundary  line  runs  N.  24°  E.  80  rd. 
to  B,  thence  N.  65°  E.  120  rd.  to  (7,  thence  S.  180  rd.  to  D, 
thence  back  to  A.     Find  also  the  length  and  bearing  of  DA. 

Ans.   99.157  acres;  i)A  =  152.07  rd.;  N.  68°  18'  23"  W. 

64.  Find  the  area  of  the  following  described  tract  of  land: 
starting  from  a  point  A.,  the  boundary  line  runs  N.  10°  E.  100  rd,, 
thence  N.  47°  E.  150  rd.,  thence  E.  40  rd.,  thence  S.  10°  W. 
100  rd.,  thence  W.  40  rd.,  thence  to  A  the  place  of  beginning. 

Ans.  81.04  acres. 


CHAPTER   III 


TRIGONOMETRIC   FUNCTIONS    OF   ANY   ANGLE.     GRAPHS 

In  §  2  the  trigonometric  functions  have  been  defined  for 
positive  acute  angles  only.  We  shall  now  extend  these  defi- 
nitions to  include  angles  of  any  size  whatever. 

19.  Axes.  Quadrants.  To  locate  an  angle  which  may  be 
either  acute  or  obtuse  it  is  convenient  to  employ  Coordinate 
Axes  as  shown  in  Fig.  19.  The 
horizontal  line  is  called  the 
X-axis,  the  vertical  is  called 
the  y-axis.  These  axes  divide 
the  plane  into  four  quadrants 
marked  I,  II,  III,  IV. 

A  positive  trigonometric  angle 
is  described  when  a  vector  OP  is 
rotated  about  0  counter-clockwise 
from  the  initial  position  OX  into 
a  terminal  position  OP ;  XOP  =  a. 
If  the  rotation  be  clockwise  about  0,  the  angle  described  is  7iega- 
tive  ;  XOP^  =  -  /3.     See  Fig.  19. 

20.  Coordinates.  Abscissa,  Ordinate.  The  position  of  a 
terminal  line  OP  (Fig.  20  a,  b,  c,  d^  is  determined  by  two 
measurements  a;,  y,  called  coordinates  of  the  point  P.  The 
x-measurement  is  called  the  abscissa  of  the  point  P,  and  is  the 
projection  of  OP  on  the  X-axis.  If  the  projection  of  OP  falls 
to  the  right  of  0  (Fig.  20  a,  c?),  the  abscissa  is  positive,  if  to 
the  left  (Fig.  20  b,  c),  the  abscissa  is  negative. 

The  y-measurement  is  called  the  ordinate  of  the  point  P,  and 
is  the  projection  of  OP  upon  the  Y-axis.     The  ordinate  is  posi- 

31 


32 


PLANE   TRIGONOMETRY 


[§§  20-21 


tive  (Fig.  20  a,  5)  when  the  projection  of  OP  falls  above  the 
horizontal,  and  is  negative  (Fig.  20  c,  c?)  when  it  falls  below  the 
horizontal. 

In  locating  a  point  in  the  respective  quadrants,  the 
coordinates  of  any  point  are  usually  written  in  brackets. 
The  symbol  (a,  5)  means  the  abscissa  x  =  a,  the  ordinate  y=h. 


■" 

- 

~ 

Y 

~ 

p 

"■ 

■- 

~ 

1 

Y 

/ 

/ 

V 

> 

^v 

/ 

r^ 

/ 

y 

71. 

s 

)-r' 

/ 

>> 

I 

\ 

s 

/ 

^. 

\ 

/ 

\ 

=  )t 

•  III 

\ 

._..  y. , 

~ 

~ 

h 

o 

t 

'  ti^ 

I 

(t-^ 

u 

'y 

_  J 

_ 

.... 

_ 

_. 

Fig.  20  a. 


Fior.  20  h. 


zr~\ 

v~                                             7"     .  .  - - 

— 

^^     '^^ 

/L 

'^                                                                               V                             V 

1 

\                               t            \ 

i:  1 i__. 

I ±    : .-_i j(_ 

'^ ^   Z7 

^_x ^     ^5^r""r zt 

%       % 

^          "^1      ^ 

^     3^ 

J>«J               -£/ 

^^ 

'^•^           ^  ^           "^^ 

^"^ 

_p 

-rt^^)                                                   X 

n.*^ 

/"j^z"^ 

V.«|r 

_Y 

"L 

_-        ± 

Fig.  20  c. 


Fig.  20  d. 


21.    Definitions  of  the  Functions.     From  the  above  figures,  (a), 
(5),  ((?),  (<?),  with  ZXOP  =  a  and  OP=r,  we  define  in  each  case: 


_V_  ordinate 
~  r~  distance  * 


C8ca  = 


cos  a 


a?  _  abscissa 
r  ~  distance ' 
,         _y  _ ordinate 
~  x~  abscissa  * 


r  _  distance 

y  ~  ordinate 

r      distance 
sec  a  =  — 


cot  a  = 


nc  ~  abscissa  * 

x     abscissa 

y  ~  ordinate  * 


§22]        TRIGONOMETRIC   FUNCTIONS   OF   ANY   ANGLE 


33 


In  this  notation  the  signs  of  the  abscissa  and  ordinate  deter- 
mine the  algebraic  signs  of  the  trigonometric  functions  for 
angles  terminating  in  the  respective  quadrants. 

22.  Laws  of  Signs.  (1)  For  all  angles  terminating  in  the  first 
quadrant,  the  functions  are  positive. 

(2)  For  angles  terminating  in  the  second  quadrant,  the  six 
functions  are  negative  except  sine  and  cosecant. 

(3)  For  angles  terminating  in  the  third  quadrant,  the  tangent 
and  cotangent  are  positive,   all  others  are  negative. 

(4)  For  angles  terminating  in  the  fourth  quadrant,  the  cosine 
and  secant  are  positive^  all  others  are  negative. 

These  laws  are  shown  in  the  following  diagram : 


Pis 

Cos 

Tan 

Cot 

Sec 

Csc 

I 

+ 

+ 

H- 

+ 

+ 

+ 

II 

+ 

- 

- 

- 

- 

+ 

III 

- 

- 

+ 

+ 

- 

- 

IV 

+ 

_ 

- 

+ 

It  should  be  noticed  that  these  general  definitions  apply 
to  angles  larger  than  360°.  For  example,  sin  400°  is  the 
sine  of  an  angle  terminating  40°  abov^e  the  initial  line ;  hence 
sin  400°  =  sin  40°.  tan  500°  =  tan  (360°  +  140°)  is  the  tangent 
of  an  angle  terminating  in  the  second  quadrant,  tan  500° 
=  tan  140°  =  -  tan  40°.     sin  (n  360°  +  «)  =  sin  a. 

The  fundamental  identities,  §  5,  hold  for  any  angle. 
sin*  a  +  cos^  «  =  1,  sin  a  csc  a  =  1, 


sec-  a  —  tan-  a  =  1, 
csc-  «  —  cot-  a  =  1, 


cos  a  sec  «  =  1, 
tan  a  cot  a  =  1. 


Examples,  i.  What  values  have  the  functions  when 
sin  ^  =  I  ? 

Solution.  The  angle  </>  may  be  constructed  in  either  of  two  positions, 
A  OP^n  the  first  quadrant,  or  the  supplement  A  OP^  in  the  second  quad- 
rant. Let  A  OPi  =  <^i,  A  OPi  =  4>r  ^^  *he  first  case,  sin  <^i  =  |,  cos  <^i  =  f, 
tan  (^j  =  I,  etc.     In  the  second  case,  sin  ^.^  =  f,  cos  ^2=  -|,  tan  <^.^=  -|,  etc. 


34  PLANE   TRIGONOMETRY  [§22-23 

2.  Find  all  the  trigonometric  functions  when  cos  <f>  =  ^. 

Solution.  Locate  angle  ^  as  a  positive  angle  in  the  first  quadrant,  and 
as  an  equal  negative  angle  in  the  fourth  quadrant.     The  angle  whose  cosine  is 

-  is  either  <^  or  —  <f>.     Then,  sin  <^  =  ±  -^— ,  cos  <!>  =  -,  tan  ^  =  ±  \/8,  cot  <l> 
=  ±  — =j  etc. 

3.  Locate  the  positive  angle  (f>,  when  P  has  the  following 
coordinates:  (a)  (4,  3);  (J)  (-4,  5);  (c)  (5,  -3);  (d)  (8, 
-1);   (e)   (-5,  -6);    (/)   (3,  -4). 

4.  Give  the  values  of  each  trigonometric  function  for  each 
angle  determined  in  Ex.  3. 

EXERCISES 

Name  the  quadrant  (or  quadrants)  in  which  (f>  terminates 
when  : 

1.  sin  (f>  =  |.  6.  tan  ^  =  —  5. 

2.  tan<^  =  4.  7.  CSC  ^  =  —  2. 

3.  cos  <^  =  f .  8.  sin  <^  =  f ,  tan  (f)  <  0. 

4.  sin  <^  =  —  1^.  9.  cos  0  =  ^,  cot  ^<  0. 

5.  cot  (f)  =  0.  10.  tan  ^  =  3,  sin  <f)  <  0. 

Express  the  following  as  functions  of  acute  angles  : 

11.  sin  (440°)  =  sin  (360°  +  80°)  =  sin  80°. 

12.  sin  370°.  13.    tan  430°.  14.    cos  (2  7r  +  20°). 

15.    tanfTT-l-— j.        16.    cot(w7r  +  — j.        17.    sinf2w7r  +  — ]• 
18.    sec  300°.  19.    tan  700°.  20.    sin  500°. 

23.  Functions  of  Negative  Angles.  To  express  the  trigo- 
nometric functions  of  a  negative  angle  in  terms  of  an  equal 
positive  angle  construct  the  negative  angle  and  an  equal  posi- 
tive angle,  Figs.  21  and  22,  In  either  drawing  Q0  acute,  or  0 
obtuse)  the  triangles  A  OP  and  A  OP^  are  equal, 

x^  =  x,     ^1  =  -  y,     rj  =  r. 


§23]       TRIGONOMETRIC   FUNCTIONS  OF   ANY   ANGLE 


35 


sin  (-  6) 


y 


cos  ( 

tan  (-  6) 


6)=  -i 
r 


1 

Vi    -y 

•^1 


X 


=  — sin  0, 
cos  9, 
=  -tane. 


CSC  ( -  6)  =  -  CSC  0, 
sec  (—  0)  =  sec  0, 
cot  (-0)  =-cot0. 


The  Laws  governing  Functions  of  Negative  Angles 
ARE:  (1)  Ay^y  trigonometric  function  of  a  negative  angle  equals 
the  same  fuyiction  of  an  equal  positive  angle,  (2)  the  sign  being 
changed  in  all  cases  except  for  the  cosine  and  secant. 

EXERCISES 

1.  Write  equivalents  of  the  following  functions  with  the 
signs  of  the  angles  changed  : 

(1)  cos  (-48°)  ;      (4)  cot  (-  87°)  ;        (7)  sin  47°  ; 

(2)  tan  (-  65°)  ;     (5)  sec  (-  75°)  ;        (8)  tan  (a  -  /3); 

(3)  sin  (-  50°)  ;      (6)  cot  (-  100°)  ;      (9)  sin  ((9  -  <f)). 

2.  Write  numerical  values  of  the  following : 

(1)  sin  (-  30°)  ;      (3)  cot  (-  60°)  ;      (5)  cos  (-  60°)  ; 

(2)  tan  (-  45°)  ;      (4)  sec  (-45°)  ;      (6)  tan  (-  90°). 

3.  Reduce  to  numerical  values  : 

(1)  sin  90°  X  sin  (-  90°)  -  tan  (-  45°). 

(2)  tan  (-  60°)  x  sin  (-  30°)  x  esc  60°. 

(3)  sin2  (-45°)  x  cos  (-  60°)  x  esc  (-  45°). 

(4)  sin2  (-  45°)  ^  cos2  (-  45°)  +  tan  (-  45°). 

(5)  sec2  (120°)  -  tan2  (120°)  +  sin  (-90°). 

(6)  cos  (-  80°)  x  sin  (-  20°)  x  sec  80°  x  esc  (-  20°). 


36 


PLANE   TRIGONOMETRY 


[§  23-24 


24.    Functions  of  90°  -  6  and  90°  +  6. 

(1)  Complemental  angles.  The  functions  of  90°  —  0  with  6 
acute,  Fig.  23  a,  or  0  obtuse,  Fig  23  6,  obey  the  same  laws  as 
have  been  developed  in  §  3. 


^'Y'     ^i(^i>yi) 


Fig.  23  a. 


Fig.  23  b. 


In  the  drawings.  Figs.  23  a,  b,  the  triangles  OP  J.  and  OP^A^ 
are  equal,  and 

^1  =  ^^   y\  =  ^^   ^1  =  ^- 
Defining  the  functions,  we  have  : 

sin  (90°  -  ^)  =  ^  =  -  =  cos  (9,      esc  (90°  -&)  =  sec  0, 
x^      r 

cos  (90°  -  6/)  =  ^  =  ^  =  sin  ^,      sec  (90°  -  6*)  =  esc  ^, 

tan  (90°  -  (9)  =  ^  =  -  =  cot  ^,      cot  (90°  -0^=  tan  ^. 
^1     y 

Law  for  Complemental  Angles.  The  functions  of  the 
complement  of  any  angle  equal  the  corresponding  cofunctions  of 
the  angle. 

(2)  Functions  of  90°  +  ^.  The  functions  of  90°  +  <9  may  be 
derived  from  the  above  functions  of  90°  —  ^  by  changing  0  into 
—  0  and  noting  the  results  of  §  23. 

sin  (90°  +  9)  =  cos  ( -  0)  =  cos  6,   esc  (90°  +  6)  =  sec  9, 
cos  (90^  +  9)=  sin  (-9)=-  sin  9,  see  (90^^  +  9)  =  -  esc  9, 
tan  (90°  +  9)  =  cot  (  - 9)  =  -  cot  9,  cot  (90°  +  9)  =  -  tan  9. 

These  results  show  that  the  same  law  holds  for  functions  of 
90°+^  as  for  90°—^,  except  the  algebraic  sign  is  changed  in  all 
cases  except  that  of  the  sine  and  cosecant. 


§25]       TRIGONOMETRIC    FUNCTIONS   OF   ANY   ANGLE  37 

25.  Functions  of  180° -6  and  180°  4- 6.  Supplemental  Angles. 
The  functions  of  the  supplement  of  an  angle  d  may  be  expressed 
in  terms  of  the  functions  of  0.  Construct  0  acute,  Fig.  24  a  ; 
also  construct  6  obtuse,  Fig.  24  b. 


Pi(a:i'2/i) 


X'-^ 


Fig.  24  a. 


Fis.  24  b. 


In  either  figure  the  triangle  OAP  equals  the  triangle  OAj^P^, 

and 

xj^=-x,   y^=y,   r^  =  r. 

Hence, 

sin  (180°  -  6)  =  ^  =  |[  =  sin  6,  esc  (180°  -  0)  =  esc  6, 

cos (180°  -  6)  =  ^  =  ^^  =  -cos  0,     sec  (180°  -  0)  =-  sec  0, 
^1       *' 

tan  (180°  -  0)  =  ^1  =  ^-  =  -tan  0,    tan  (180°  -  0)  =  -  tan  0. 

Laws  for  Supplemental  Angles.  Any  trigonometrie 
function  of  an  angle  equals  the  same  function  of  its  supplement^ 
the  algebraic  sign  being  changed  in  all  cases  except  the  sine  and 
cosecant. 

The  functions  of  180°  +  6  may  be  obtained  from  those  of 
180°  —  6  by  changing  ^  to  —6  in  the  above  formulas.     We  find  : 

sin  (180°  +  0)  =  sin  (-  0)  =  -  sin  0,  esc  (180°  +  0)  =  -  esc  0, 
cos  (180°  +  0)  =  -  cos  ( -  0)  =  -  cos  0,  see  (180°  +  0)  =-  sec  0, 
tan  (180°  +  0)  =  -  tan  (-  0)  =  tan  0,      cot  (180°  +  0)  =  cot  0. 

Law.  Any  trigonometric  function  of  180°  ±  9  equals  the  same 
function  of  the  angle  6,  regard  being  had  for  the  algebraic  sign. 


374041 


38 


PLANE   TRIGONOMETRY 


[§25 


EXERCISES 

1.    Fill  the  blanks  with  the  proper  function  of  the  supplement 


of  each  angle : 

(1)  sin  150°  =  sin  30° 

(2)  tan  97°  20'  = 

(3)  cos  160°  40'  = 

(4)  cot  175°  10'  = 
'  (5)  sec  120°  10'  = 


(6)  CSC  100°  20'  = 

(7)  tan  (90°  +  </))  = 

(8)  sin  (90°  -  </))  = 

(9)  cos(45°-<^)  = 
(10)  cot  (60°  +  <^)  = 


2.  By  taking  supplements  of  0°,  30°,  45°,  and  60°  find  the 
trigonometric  functions  of  180°,  150°,  135°,  and  120°.  Fill  the 
blanks  in  the  following  table : 


<!>  = 

0° 

30° 

45° 

60° 

90° 

120° 

135° 

150° 

180° 

sin  <^ 

1 

n^^ 

^J- 

1 

iU 

:1ol 

V 

cos  <^ 

1        i             \-        •■ 

"-fy 

a- 

■M' 

MM 

\k. 

'  1 

tan  <f) 

lc.'\^ 

1 

l.n 

r^ 

At 

5V 

c 

cot  <^ 

i'\^ 

/ 

4T> 

O 

•■^77 

sec  <^ 

, 

■  ■ 

n^ 

cm>. 

-U^/ 

' 

'■  1 

CSC  <f> 

J     1     -y 

l,iS.' 

r 

■\y. 

3.    Find  numerical  values  of  the  following : 

(1)  sin  120°  X  sin  60°. 

(2)  tan  45°  x  cot  12°  x  cos  90°. 

(3)  5  X  cos  135°  X  sin  90^  x  cos  180°. 

(4)  'tan  135°  x  cot  130°  --sin  60°. 

(5)  tan  150°  x  cos  150°  ^  sin  30°. 

(6)  2  sin  120°  +  cot  150°. 

(7)  tan  135°  +  cot  45°  -  cos  180°. 

(8)  cos  30°  +  cos  150°  +  tan  60°  +  tan  120°. 

(9)  (tan  120°-  tan  135°)  x  (tan  120°  +  tan  135°). 
(10)  sec  (180°  -  ^)  X  cos  ^  X  tan  (180°  -  a)  x  cot  a. 

4.    From  proper  drawings  obtain  values  of  the  functions  of 
180°  +  ^  in  terms  of  the  functions  of  0. 


§26]       TRIGONOMETRIC   FUNCTIONS   OF   ANY   ANGLE  39 

5.  Verify  sin  210°  =  -  -| ;  cos  225°  =  -  l  V2 ;  tan  225°  =  1; 
tan  240°  =  V3.      [Note.     Construct  drawings.] 

6.  By  adding  180°  to  the  angle  in  the  functions  of  90°  —  6 
derive  the  functions  of  270°  —  ^.  Verify  the  results  by  con- 
structing proper  drawings. 

7.  Find  the  value  of  ^^^  i^^°  +  ^')  +      ^an  (- «)      ^ 

sin(-«)  tan  (180°  + a) 

8.  Find  the  value  Of  5HiO^^:i^x^^^?^^±^. 

cos  (90°  +  <^)        cot  (90°  +  6) 

9.  What  sign  has  sin  x  +  cos  x  for  the  following  values  of  x : 
(1)  a;  =  90°;  (2)  a;  =120°;         (3)  a;  =  135°; 

(4)  a;  =  210°;  (5)  a:  =300°? 

10.    Find  all  angles  less  than  360°  which  satisfy : 
(1)  tan^  =  -l;     (2)  sin(9=iV3;     (3)  cos^  =  -i. 

26.    Line-values  of  the  Functions. 

(1)  Acute  angles.     Let  the  angle  6  be  constructed  at  the  cen- 
tre of  a  circle  whose  radius  is  unity,  Fig.  25.     Then  arc  AB=6. 


COT    0 


y^ 

\ 

i»x 

/ 

•\^> 

[T 

/                  E 

z' 

y 

s, 

<*» 

\ 

/ 

90VS^ 

/ 

<*> 

z 

\ 

< 

A' 

10 

1 

A 

0 

cos 

c 

/ 

y 

T' 


(7B  =  sin6',  OT'=csc^, 
0(7=  cos  ^,  OT  =  sec^, 
^r  =  tan^,i)T'  =  cot^, 
CA  =  vers  B,  ED = covrs  B. 

These  lines  represent 
graphically  the  values  of 
the  trigonometric  func- 
tions when  the  radius  of 
the  circle  is  taken  as 
unity. 

It  should  be  noticed 
that  trigonometric  func- 
tions considered  as  line-values  may  be  described  as  follows : 
(1)  The  sine  of  an  angle  is  the  length  of  the  perpendicular  let 
fall  from  the  terminal  end  of  the  arc  (-B)  upon  the  diameter 
through  the  initial  end  (J.).     (2)   The  cosine  of  an  angle  is  the 


Fig.  25. 


40 


PLANE   TRIGONOMETRY 


[§26 


part  of  the  radius  {00}  cut  off  hy  the  foot  of  the  perpendicular 
{BO}.  (3)  The  tangent  of  an  angle  is  the  geometric  tangent 
erected  at  the  initial  extremity  {A)  of  the  arc  and  terminated  hy 
the  diameter  produced  through  the  terminal  extremity  (5)  of  the 
arc.  (4)  The  secant  is  the  line  from  0  to  the  extremity  of  the  tan- 
gent {OT).  (5)  The  cotangent  is  the  geometric  tangent  erected 
at  the  90°  point  (D)  of  the  circle^  and  terminated  by  the  terminal 
radius  produced  to  T\  {DT'}.  (6)  The  cosecant  is  the  length 
of  the  line  from  the  centre  of  the  circle  to  the  extremity  of  the 
cotangent  {OT'}. 

Oonvention  of  signs.  In  the  above  definitions  the  sines  and 
tangents  are  verticals,  and  are  taken  positive  when  draivn  up- 
ward., negative  when  drawn  downward.  The  cosines  and  cotan- 
gents are  horizontals,  and  are  positive  when  drawn  to  the  right, 
negative  when  drawn  to  the  left.  The  secants  and  cosecants  are 
positive  when  drawn  from  0  along  the  terminal  boundary  OB, 
negative  when  drawn  from  0  backward,  as  OT  in  Fig.  26  (a). 

The  older  text-books  defined  the  trigonometric  functions 
from  this  line-value  standpoint,  but  at  the  present  time  the 
definitions  are  usually  given,  as  in  §  2  and  §  21,  from  the  ratio 
standpoint.     It  is  often  convenient  to  use  the  line-values. 

(2)  Angles  larger  than  90°.  Drawings  are  here  inserted  to 
show  the  line-values  of  the  functions  for  angles  larger  than  90°, 
Figs.  26  (a),  (J),  (c).     In  each  case  0T=  sec  6,  OT'  =  esc  6. 


V  COT    Oi        D 


D        COT   02       T' 


(6) 

Fig.  26  (a),  (6),  (c). 


r  COT  63 

D 

1   V 

1                 7  ^ 

1                 /     \ 

5^ 

\        ^ 

Vos^ 

% 

M 


These  drawings  show  clearly  that  the  fundamental  relations, 
§  5,  hold  whatever  the  angle  may  be. 

The  following  table  shows  the  signs  of  the  functions  of  B,  0^, 
6^,  ^3,  shown  in  Figs.  25,  26  (a),  (6),  (<?),  respectively. 


§  26]       TRIGONOMETRIC   FUNCTIONS   OF   ANY    ANGLE 


41 


Sin 

Cos 

Tan 

Cot 

Sec 

Cso 

e 

+ 

+ 

+ 

+ 

+ 

+ 

e. 

+ 

- 

- 

- 

- 

+ 

61 

- 

- 

+ 

+ 

- 

- 

e. 

- 

+ 

- 

- 

+ 

- 

It  should  be  remarked  that  the  functions  of  an  angle  between 
270°  and  360°  are  the  same  as  those  of  a  negative  acute  angle, 
and  that  the  functions  of  an  angle  between  180°  and  270°  are 
the  same  as  those  of  a  negative  obtuse  angle. 

EXERCISES 
Draw  figures  and  show  : 

1.  sin  70°  =  cos  20°  =  sin  110°. 

2.  sin  120°  =  cos  30°  =  -  sin  (  -  60°). 

3.  sin  117°  =  cos  (-27°)  =  cos  27°. 

4.  cos  300°  =  cos  (  -  60°)  =  sin  30°. 

5.  sin  220°  =  sin  (  -  40°)  =  -  sin  40°. 

6.  cos  195°  =  -  cos  (  -  1 5°)  =  -  cos  15°. 

7.  sin (-10°)=  sin  190°  = -sin  10°. 

8.  sin  60°  =  -  cos  150°  =  cos  30°. 

9.  tan  120°  =  -  tan  60°  =  tan  (  -  60°). 
10.   cot  165°  =  -  cot  15°  =  -  tan  75°. 


If  ^  +  5  + C=  180°,  show 

A       .    B+C 
11.    cos  —  =  sin  — 


,,      .    B           A+C 
12.   sin  —  =  cos  — 

9  9 


13.  sin  A  =  sin  (5  +  C). 

14.  cos5  =  — cos(yi+ (7). 

15.  tan  A  =  —  tan  (5  +■  C). 

16.  sinJ.=  -sin(2J.-f  5+C). 


17.  cos^  =  -cos(2^  +  5+ C). 

'B-C\       .    /A  +  2B\       .    M+2 

18.  cos  ( z —  I  =  Sin  f  —^ 1  =  sin  ' 


42 


PLANE   TRIGONOMETRY 


[§27 


,^      -    ;A-B\               f2A+C\            fC+2B\ 
19.    sin  (  — - —  )  =  —  cos  I 1  =  cos  [ J- 


20.   COS  B  =  sin 


2     7  V       2 

A  +  SB  +  cr 


fA  +  SB  +  OX 


Change  each  of  the  following  to  functions  of  positive  acute 
angles : 

21.  sin  340°.  24.   cot (- 160°).        27.    sec  (- 200°). 

22.  tan  540°.  25.    sin  260°.  28.    tan  (-280°). 

23.  cos  450°.  26.   cot  300°.  29.   sin(-500°). 


30.   cos  (-490°). 


31.   sin(-40°)xtanl87°. 


'6   _B 


27.   Graphs  of  the  Trigonometric  Functions.     The  line-values 
of  the  functions  furnish  a  means  for  constructing  graphs  which 
show  to  the  eye  the  numerical  values  of 
sin  a;,  cos  x,  tan  x  for  any  angle. 

(1)  Graph  of  sin  x.  Let  us  take  a  unit 
circle,  Fig.  27,  and,  beginning  at  A,  locate 
points  5p  B^,  ^g,  •••  at  equal  intervals, 
say  18°,  along  the  circumference.  Then 
draw  perpendiculars  B^C^,  -^2^2'  •■"• 
These  lines  are  equal  to  the  sines  of  the 
respective  angles  subtended  by  arcs  AB^, 
AB^,  •••.  Now,  place  the  point  A  at  0,  Fig.  28  («),  and 
straighten  the  circumference  along  the  line  OX,  the  points 
B^,  B^,  jBg,  •••  falling  at  equal  intervals  along  OX  as  indicated. 
Erect  perpendiculars  (+  or  — )  equal  to  B^C^,  B^C^,  -^3^31  •"■> 
respectively.  These  lines  are  the  sines  of  the  angles  whose  arc 
measures  are  AB^,  AB^,  AB^,  •••.  In  Figs.  28  (a)  and  (5)  the 
scale  is  ^  of  that  used  in  Fig.  27. 

If  a  smooth  curve  be  drawn  through  Cj,  O^,  Cg,  •••,  we  have 
a  locus  known  as  the  sine  curve,  or  the  graph  of  sin  x. 


Fig.  27. 


Fig.  28  (a).     SINE  curve. 


§27]       TRIGONOMETRIC   FUNCTIONS   OF   ANY   ANGLE 


43 


If  the  circumference  be  laid  off  to  the  left  of  0,  we  have  the 
part  of  the  graph  shown  on  OX' . 

(2)  G-raph  of  cos  x.  Taking  the  unit  circle  as  before,  Fig. 
27,  dividing  its  arc  into  equal  intervals,  laying  the  circumfer- 
ence upon  OJT,  and  erecting  perpendiculars  at  (9,  B^^  B^^  B^,  .•• 
equal  in  length  to  OA,  OC^  OC^,  •••,  we  have  lines  which  equal 
the  cosines  of  the  angles  subtended  by  arcs  AB^,  AB^,  •••,  re- 
spectively. Now,  connect  the  points  A,  C^,  C^-,  •••  by  a  smooth 
curve,  and  we  have  the  graph  of  cos  x,  or  the  cosine  curve. 
Fig.  28  (6). 


Fig.  28  (6).       COSINE    CURVE. 


(3)  Graph  of  tan  x  and  cot  x. 
on  OX,  Fig.  28  (c),  ^ve 
erect  perpendiculars  ( -}- 
or  — )  equal  to  the 
lengths  of  the  tangents 
at  A  of  the  arcs  AB^, 
AB^,  ABq,  •••,  the  points 
7\,  2Tj,  •••  lie  upon  a 
curve  known  as  the 
tangent  curve,  or  the 
graph  of  the  tan  x.  This 
curve  consists  of  a  series 
of  parallel  graphs,  each 
extending  to  infinity  at 
90°,  270°,  540°,  .-. 

The  graph  of  cot  x 
may  be  constructed  in  a 
similar  manner.  The 
graphs  of  tan  x  and  cot  x 
are  shown  in  Fig.  28  (c). 


If  at  the  points  B^,  B^,  B^,  ••• 


Fig.  28  (c). 


r< 


l<^ 


CHAPTER   IV 
MEASUREMENT  OF  ANGLES 

28.  Units  of  Measure.  In  the  measurement  of  angles,  two 
units  are  in  use,  the  degree  and  the  radian. 

(1)  Sexagesimal  system.  The  degree  *  (marked  by  °)  is  the 
-^^  of  a  right  angle.  It  is  the  unit  of  degree  measure  or  the 
sexagesimal  measure.  The  degree  is  divided  into  60  minutes 
(marked  60'),  and  the  minute  into  60  seconds  (marked  60"). 

48  degrees  35  minutes  24  seconds  =  48°  35'  24". 

Seconds  are  frequently  written  as  decimals  of  a  minute,  and 
minutes  and  seconds  may  be  written  as  decimals  of  a  degree. 

45°  21'  36"  =  45°  21.6'  =  45.36°. 

Degree  measure  is  used  in  most  of  the  practical  calculations 
of  astronomers,  engineers,  and  surveyors. 

(2)  Radian  measure  (^Circular 
measure^.  In  this  system  the  unit 
of  measure  is  the  radian^  the  angle 
subtended  at  the  center  of  a  circle  hy 
an  arc  equal  to  the  radius. 

In    Fig.    29,    the    arc    BC  =  OC, 
hence    the    angle    COB=  1    radian. 
Circular  measure  is  used  almost  ex- 
clusively in  the  theoretical  work  of 
Fig.  29.  higher  mathematics. 

29.  Relations  between  Degree  and  Radian  Measure.  The  two 
systems  of  measurement  may  be  compared.  If  (7  =  circumfer- 
ence, R  =  radius,  we  have 

e=  2  7ri2,  TT  =  3.14159265 

*  The  degree  could  as  well  be  defined  as  the  in  of  an  angle  of  an  equilateral  triangle. 
In  their  efforts  to  determine  direction,  it  is  thought  the  ancient  Babylonians  used  as 
a  unit  of  measure  the  angle  of  an  equilateral  triangle ;  this  angle  was  at  first  divided 
into  10  equal  parts,  and  later  into  60,  which  coincides  with  our  sexagesimal  system. 

44 


§  29]  MEASUREMENT   OF   ANGLES  45 

A  circumference  subtends  360°  at  the  centre ;  if  the  radius  be 
taken  as  the  unit  of  measure  (called  a  radian),  we  have  360°  in 
angular  measure  equal  to  2  tt  radians  in  circular  measure. 

360°  =  2  TT  radians, 
180°  =  IT  radians, 

1°  =  —radian  =  0.0174o3292ol99-  •  radian, 
180 

1'  =0.0002908882086-    radian, 
1"  =  0.0000048481368     radian. 
Conversely, 

1  radian  =  ^^  =  57. 29577° =  57°  17'  448". 

TT 

Conversion  Rules.     (1)   To  reduce  degrees  to  radians  mul- 
tiply the  number  of  degrees  by  — -  =  .01745,  and  mark  the  result 

radians.     (2)   To  reduce  radians  to  degrees  multiply  the  number 

180 
of  radians  by  ■ — '—  =  57.295,  and  mark  the  product  degrees. 

IT 

For  approximations  take  tt  =  ^. 

Examples,    l.  Convert  63°  30'  to  radians. 
63.5  X  0.01745  =  1.099  radians. 

2.  Convert  148°  20'  24"  to  radians. 

148°  20'  24"  =  148.34°. 
148.34  X  0.01745  =  2.588  radians. 

3.  Change  2.5  radians  to  degrees. 

2-.5  X  57.295  =  143.239  degrees. 


EXERCISES 

TT         TT         TT         7r 


/-^.           t                                                n    II           II           II           II          ^  TT  O  TT 

^.    (jive  degree  measure  or  —  :    -  ;   .— ;    — ;    —— ;  — —  • 

2       o       4       b        o  4 

2.  Change  from  degrees  to  radians :  18°;  124°;  290°;  30°; 
500°  45'. 

3.  Change  from  radian  measure  to  degrees:  1.5;  |;  1.65; 
2.75;    -0.4;  5|. 


46 


PLANE   TRIGONOMETRY 


[§§  29-30 


4.  Find  the  number  of  degrees  in  :   (a)  1  +  ^ ;   (i)  1  —  ^ ; 
(c)  140°  -  11;  id)  30°  +  2  tt;  (^  '^0°-  ^ ;    (/)   B  -  ^  tt. 

5.  What  does  each  of  the  following  symbols  mean? 
Note,     n  is  any  integer. 

(a)    ±7r;     (6)w7r;     ((?)2w7r;     ((f)  w J;     (e)   (2w  +  1)|; 

(/)   (2n-l)|;   (^)  7^7r±|. 

6.  Name  the  quadrant  in  which  each  of  the  following  angles 
terminates : 

(«)  TT-l;    (6)  7r  +  48°;    (c)  240°  +  2 ;    (<?)  W7r±^,   when 
w=0,  1,  2,  3,  4. 

7.  Determine  the  smallest  positive  angle  which  has  the  same 
terminal  boundary  as  : 

(a)  440° ;    (5)  660° ;    (c)  378° ;    (<f)   -  200° ;    (e)    -  300° ; 
(/)  7r-40°;   (^)   27r  +  120°. 

30.    The  Length  of  Any  Arc.     In  geometry  it  is  shown  that 

in  unequal  circles  arcs  subtending 
equal  angles  are  to  each  other  as 
the  corresponding  radii.  Thus,  in 
the  figure  arc  PQ  :  arc  AB  ::  OP 
:  OA.  Hence,  if  OA  be  taken  as  a 
unit,  the  arc  AB  is  the  radian 
measure  of  the  angle  A  OB ;  that  is, 
AB  =  e;    and  PQ  =  e-  OP=0-B. 

Theorem.  The  length  of  any 
arc  equals  the  radius  multiplied  by 
the  radian  measure  of  the  angle 
subtended  by  the  arc. 


Fig.  30. 


Examples,    i. 
of  radius  27  ft. 
Solution. 


Find  the  length  of  a  20°  arc  upon  a  circle 


20° 


Arc  of  20° 
Note.     Take  nr  =  V- 


20  x-^?L_=^  radian. 
18U      9 

5:x  27  ft.  =  3  TT  ft.  =  9.42  ft. 


30-31] 


MEASUREMENT   OF   ANGLES 


47 


2.  Find  the  length  of  an  arc  subtending  135°  upon  a  circle 
whose  radius  is  16  in. 

Solution.  135°  =  135  x  -^  =  -  tt  radians. 

180      4 

Arc  of  135°  =  f  TT  X  16  =  12  tt  in.  =  tt  ft. 

3.  Find  the  angle  subtended  at  centre  by  a  6  ft.  arc  upon 
a  circle  whose  radius  is  15  ft. 


SOLUTIOX. 


Arc  =  0-R. 

arc       6 


e  = 


R 


-  radian. 
15      5 


Hence,  0  expressed  in  degrees  is 

I  X  57.29°  =  22.91°  =  22°  54'  36". 

31.    Segment  and  Sector  Areas.     The  area  of  a  sector  of  a  cir- 
cle, AOB,  equals  arc 
AB  multiplied  by  one 
half    of    the    radius, 
Fig.  31. 

Sector  AOB  = 
|arc^^xOJ5  =  |i^^. 

Area  of  triangle  ^05 
=  jAB  X  OB 
=  ^AJEx  OB 
=  17^2  sin  <9. 

Area  of  shaded  segment 


2 


(9 -sine). 


Note.     The  angle  $  is  to  be  expressed  in  radian  measure. 

EXERCISES 

M  1.    Find  the  length  of  an  arc  of  63°  on  a  circle  of  20-in. 

radius.  Ans.    22  in. 

Note.     In  this  list  of  exercises,  take  tt  =  ^/,  except  otherwise  indicated. 

V    2.    Through  what  distance  does   the   extremity  of   a   7-in. 
minute  hand  of  a  clock  move  in  10  min.  ?  Ans.    3|  in. 


48  PLANE   TRIGONOMETRY  [§31 

3.  Find  the  radius  of  a  circle  in  which  a  10-ft.  arc  subtends 
an  angle  of  2  radians  at  the  centre.  Ans.    5  ft. 

4.  Find  the  angle  at  the  centre  of  a  circle,  radius  24  in., 
which  is  subtended  by  an  11-in.  arc.  Ans.    26°  15'. 

5.  Find  the  area  of  a  24°  sector  whose  subtending  arc  is 
8  ft.  Ans.    76.36  sq.  ft. 

^•j  6.  In  a  circle  whose  radius  is  10  ft.,  a  chord  is  6  ft.  from 
the  centre.  Find  the  area  of  the  smaller  segment  cut  off  by 
the  chord.  Ans.    44.73  sq.  ft. 

Note.     Take  radian  measure  from  Table  IV  in  Ex.  6,  7, 8,  9, 10, 15, 16, 17. 

"^  7.  A  horizontal  oil  tank  whose  length  is  30  ft.  and  radius 
4  ft.  is  filled  to  a  depth  of  15  in.  Find  the  number  of  gallons 
of  oil  in  the  tank.     (231  cu.  in.  =  1  gal.)         Ans.    1125.6  gal. 

^  8.  Find  the  length  of  a  belt  stretched  around  two  pulleys 
whose  radii  are  5  ft.  and  1|^  ft.,  respectively,  the  distance 
between  the  centres  of  the  pulleys  being  20  ft.  Also  find  the 
length  of  the  belt  when  crossed. 

Ans.    (a)  61.03  ft. ;   (K)  62.55  ft. 

9.    Find  the  distance  at  which  a  foot  ruler  must  be  held  so 
that  its  length  will  subtend  one  degree  at  the  eye. 

Ans.    57.29  ft. 

10.  Find  the  length  of  a  degree  on  a  circle  whose  radius  is 
3956  mi.  Ans.    69.045  mi. 

11.  What  is  the  length  of  a  degree  of  longitude  on  the 
39th  parallel  of  latitude,  if  one  degree  on  the  equator  is  69.045 
mi.,  the  earth's  radius  being  3956  mi.  Ans.    53.65  mi. 

12.  If  m,  be  the  length  of  one  degree  upon  the  equator,  show 
tliat  tlie  length  of  one  degree  upon  any  parallel  of  latitude  is 
m  X  cos  a,  when  a  is  the  latitude  of  the  parallel. 

13.  In  radian  measure  two  angles  of  a  triangle  are  ^  and  |. 
Find  the  third  angle  in  degrees  (tt  =  -^^).  Ans.    138°. 

14.  The  end  of  a  30-in.  pendulum  swings  through  a  3-in. 
arc.     Find  the  angle  through  which  it  swings.      Ans.    5°  43.8'. 


§31]  MEASUREMENT   OF   ANGLES  49 

15.  If  the  diametei-  of  the  moon  be  2163  mi.,  find  tlie 
moon's  distance  from  the  earth,  assuming  that  its  diameter  sub- 
tends 31'  7"  at  the  eye.     See  §  29.  Atis.    238,900  mi. 

16.  If  the  sun's  apparent  diameter  subtends  an  angle  of  32'  4" 
at  the  eye,  find  its  diameter,  assuming  the  sun's  distance  from 
tlie  earth  to  be  92,897,000  miles.  Ans.    866,500  mi. 

17.  If  the  earth's  equatorial  radius  (3963  mi.)  subtends  8.8" 
at  the  sun  (the  sun's  parallax),  find  the  distance  of  the  sun 
from  the  earth.  Ans.    92,890,000  mi. 


CHAPTER   V 


FUNCTIONS   OF   TWO   ANGLES.     MULTIPLE   ANGLES 

We  next  develop  the  trigonometric  functions  of  the  sum  and 
difference  of  two  angles. 

32.  To  develop  sin  (a  +  P)  and  cos  (a+P).  The  angles  a 
and  /3  may  be  both  acute  and  such  that 
their  sum  is  less  than  90°,  as  in  Fig.  32  ; 
or  they  may  be  such  that  their  sum  is 
greater  than  90°  and  smaller  than  180°, 
as  shown  in  Fig.  33.  In  each  con- 
struction, /3  is  made  to  join  a,  the  line 
PQ  being  erected  perpendicular  to  OP 
and  QN  (^QN'),  drawn  perpendicular 
to  OM  at  iV,  or  to  OM  produced  nega- 
tively at  i\r. 

Then  defining  sin  (a  +  /3)  from  the  figure,  we  have 

^   ^""^      OQ  OQ 

^MP^qP.QR^QP 
OQ       OP'^  OQ      QP 

=  sin  a  cos  yS  +  cos  a  sin  yS. 

Similarly, 

cos  (a  +  yS)  = 

QP 


Fig.  32. 


ON 

OQ 

OM 

OQ 


OP 


RP 

OQ  "^  QP 


=  cos  a  cos  —  sin  a  sin  /3. 
60 


§33]  FUNCTIOXS   OF   TWO   ANGLES  51 

33.    To  develop  tan  (a+  P).    (1)  We  may  derive  tan  («+  /S) 
from  the  drawing,  thus  : 

PM^QR 


^       ^      OJY     OM-PR        .      PR 

OM 

tan  a  +  ^ 

OP         tan  a  -f  tan  yS 


^  _  PM    PQ     1  -  tan  a  tan  fi 
OM-  OP 

Note.     The  triangles  OMP  and  QRP  are  similar. 

(2)  We   may  derive   tan  (a  +  yS)  analytically   by   dividing 
sin  (a  4- 1^')  by  cos  (a  +  yS). 

tan  Ca  +  yS^  =  sin  (cc  +  ^)  _  sin  «  cos  y8  +  cos  a  sin  y8  _ 
cos  («  +  /3)      cos  a  cos  yS  —  sin  a  sin  yS 

Dividing  both  numerator  and  denominator  by  cos  a  cos  y8,  we 
find 

t.an(«  +  yQ)=  tan«  +  tan/3  ^ 
1  —  tan  « tan  /3 

By  taking  the  reciprocals  of   tan  (a  -f-  yS),  tan  a,  tan  yS,  we 
have 

cot  a  cot  /S  —  1 


cot  (a  4-  y8)  = 


cot  a  +  cot  yS 


Since  the  sin  C  —  ^)  =  —  sin  ^,  cos  (—  ^)  =  cqs  6,  and  tan  (  —  ^) 
=  —  tan  ^,  we  have,  on  changing  y8  into  —  yS  in  each  formula 
above : 

sin  (a  —  ^^=  sin  a  cos  y8  —  cos  a  sin  /8, 
cos  (a  —  /3)  =  cos  a  cos  yS  +  sin  a  sin  yS, 
tan  a  —  tan  y8 


tan  (a  —  /3)  = 


1  +  tan  a  tan  yS 


Note.  The  above  formulas  have  been  derived  from  figures  in  which 
a  +  yS  is  either  acute,  or  lies  between  90*^  and  180°.  The  same  results  are 
true  for  any  values  of  a,  /3,  but  we  shall  not  introduce  any  proof  of  this 
fact  here. 


52  PLANE   TRIGONOMETRY  [§34 

34.    Important  Formulas.     Collecting  the  results  of  the  above 
demonstrations,  we  have  the  fundamental  formulas: 

(1)  8in(a  + P)  =  sin  a  cos  p  +  cosasin  p, 

(2)  cos  (a  +  P)  =  cos  a  cos  p  —  sin  a  sin  p, 

(3)  sin  (a— P)  =  sin  a  cos  P  — cos  a  sin  p, 

(4)  cos  Ca  —  P)  =  cos  a  cos  p  +  sin  a  sin  p, 

/c\  *     ^     ,  On      tan  a  +  tan  P 

(5)  tan(a+P)  =  -- -~^, 

1  —  tan  a  tan  p 

/«\  *     /       On      tana  — tan  B 

(6)  tan(a-P)  = ^, 

1  +  tan  a  tan  p 

i^7\  «  *  /     ,  On      ±  cot  a  cot  P  -  1 

(7)  cot(a±p)=— — — - — r  — 
^  cot  a  ±  cot  p 

yL.J^J^^^^'"^^  EXERCISES 

1.  Show  sin  75°  =  sin  (45°  +  30°)  =  i( V6  +  V2). 

Solution,     sin  (45°  +  30°)  =  sin  45°  cos  30°  +  cos  45°  sin  30° 

^  V2  _  V3  ,   V2  _  1 
~   2    '    2         2    *  2 

Hence,  sin  75°  =  cos  15°  =  i(  v/6  +  V2). 

2.  Show  sin  15°  =  i(  V6  -  V2)  =  cos  75°. 

3.  Show  tan  75°  =  2  -f  V3  =  cot  15°. 

4.  Show  tan  15°  =  2  -  VB  =  cot  75°. 

V2 

5.  Show  sin  (45°  4- a)  = (cos  a  +  sin  a). 

Note.     Apply  formula  (1). 

6.  Prove  cos  (30°  —  a)  =  i  ( V3  cos  a  +  sin  a). 

7.  Provetan(45°  +  a:)=^i^^5^=cot(45°-a;). 

1  —  tan  X 

V3  1 

8.  Prove  -—  cos  ic  —  -  sin  x  =  sin  (60°  —  x). 

9.  Prove  -cosa;  +  -— sinx  =  cos  (60°  — a;). 

A  A 

,  r>       r)  i       racvi  n         V3  —  tan  X 

10.    Prove  tan(bO°  —  a;)  = 


1  +  V3  tan  X 


§§34-35]  FUNCTIONS  OF   TWO   ANGLES  63 

11.  Derive    the     formula    sin  («  -f  /3  +  7)  =  sin  a  cos  ^  cos  7 

+  cos  a  sin  yS  cos  7  +  cos  a  cos  /3  sin  7  —  sin  a  sin  /3  sin  7. 
Suggestion.     Write  sin  (a  +  yg  +  y)  =  sin  («+[/?  +  y]),  and  apply  (1). 

12.  Derive  cos(a  +  /3  +  7)=  cos  «  cos /S  cos  7—  cos  «  sin  yS sin  7 
—  cos  yS  sin  a  sin  7  —  cos  7  sin  a  sin  yS. 

13.  Derive  tan  (x  +  ^  +  2) 

_  tan  X  +  tan  y  +  tan  g  —  tan  x  tan  3/  tan  g 
1  —  tan  a;  tan  y  —  tan  ?/  tan  z  —  tan  g  tan  x 

14.  If  sin  2;  =  0.5  and  cos  y  =  0.6,  find  the  value  of  sin  (x+y^, 
and  cos  (x+  y'). 

Suggestion.     Construct  angles  x,  y,  and  expand  sin  (x  +  y),  then  substi- 
tute.    Likewise  with  cos  (x  +  y). 

sin  (x  +  ?/)  =  sin  x  cos  2/  +  cos  x  sin  y 

=  0.5  X  0.6  +  0.5 V3  X  0.8  =  —  (  V3  +  4). 
10  ^  ^ 

cos  (j;  +  ?/)  =  cos  X  cos  ^/  —  sin  x  sin  z/. 

=  0.5 V3  X  0.6  -  0.5  X  0.8  =  —  (V3  -  4). 
10  ^      ^ 

15.  If    sin  a:  =  0.8,  tan  ?/ =  1,    what  value    has    sin  (a;  —  ?/)? 
tan(a:  + ?/)? 

16.  When    tana;  =  V3,    cos?/ =0.5,   find    tan  (re  — y);    also 
cos  (x  —  y^. 

17.  When  cos  a;=l,  tan  ?/  =  !,  find  sin  (ic  +  ^);  also  cot  (a; +y). 

18.  Show  cos  (x  +  y^=  0,  when  tan  a;  =  0.5  and  cot  y  =  0.5. 

19.  Show  sin  (x-{-  y')  =  1,  when  sin  x  =  cos  y. 

20.  Show  tan  (a;  +  ?/)  =  00  ,  when  cos  a;  =  sin  y,  or  tan  x  =  cot  ^. 

FUNCTIONS   OF   MULTIPLE   AND    SUB-MULTIPLE   ANGLES 

The  formulas  enumerated  in  §  34  give  rise  to  new  and  im- 
portant results  by  making  angle  /3  equal  to  angle  a. 

35.    Functions  of  2  a.     In  the  formulas, 

(1)      sin  (a  -I-  yS)  =  sin  a  cos  /3  -|-  cos  a  sin  y8, 

(3)      cos  (a  +  y8)  =  cos  a  cos  /3  —  sin  a  sin  /3, 

/'t^N      4.      /^     .   /o\       fan  a  +  tan  /3 

(5)      tan(a  +  ^)=- ,         ,       r,^ 

1  —  tan  a  tan  p 

\ 
\ 


64  PLANE   TRIGONOMETRY  [§§35-37 

let  /S  =  a,  and  we  derive  : 

(7)  sin  2  a  =  2  sin  a  cos  a, 

(8)  cos  2  a  =  cos^  a  —  sin^  a  =  2  cos^  a—  1  =  1  —  2  sin^  a, 

/'Q\  .      „  2tana 

(9)  tan  2  a  = 


1  -  tan^  a 

36.    Functions  of  3  a. 

Let  yS  =  2  a  in  (1),  (3),  (5)  above,  and  employ  results  of 
(7),  (8),  (9),  and  we  obtain: 

(10)  sin  3  a  =  3  sin  a  —  4  sin''  a, 

(11)  cos  3  a  =  4  cos^  a  —  3  cos  a, 

3  tan  a  —  tan^  a 


(12)  tan3a  = 


3  tan^  a 


37.   Half-angle  Formulas.     In  (7),  (8),  (9)  §  35,  let  «  =  |, 
and  we  find : 

(13)  sin  a;  =  2  sin -cos-, 

(14)  cos  X  =  cos^ sin^  -  =  2  cos^  -  —  1  =  1  —  2  sin^  -, 

^     ^  2  2  2  2 


(15)  tan  X  = 


2  tan  ? 


l-tan2^ 


Solving  each  of  the  latter  two  values  of  cos  x  for  sin  -  and 
cos-,  we  have  the  important  formulas: 


(16)  «inf=V- 


cos  35 


(17)  cosf:.^ 

and  dividing  sin-  by  cos-. 


X ^  1  +  cos  iC 


(18)  ten|=V[ 


—  cos  ic        sin  05         l-cosaj. 


+  cos  35     1  +  cos  a?        sin  x 
(16),  (17),  (18)  are  known  as  the  half-angle  formulas. 


§37]  FUNCTIONS   OF   TWO   ANGLES  65 

j  EXERCISES 

1.  If  sin  a;  =  |,  find  sin  2  x ;  also  tan  2  x. 

2.  If  cos  X  =  ^^g,  find  tan  2  x ;  also  cos  2  x. 

3.  If  CSC  a;  =  2,  find  sin  2  a: ;  also  cos  2  a;. 

4.  P'ind  sin  x.  if  cos  -  =  0.3. 

2 

a;       5 

5.  Find  tan  x.  if  sin  -  =  -— . 

2      13 

V3 

6.  Find  sin  x.  if  cos  2  a;  =  — —  • 

2 

*7.  Find  tan x,  if  sin  2x  =  ^|-. 

"^  8.  Find  tan  60°,  knowing  tan  30°  =  I V3. 

9.  Find  tan  53°  8',  if  tan  26°  34'  =  0.5. 

10.  P'ind  cos  2  x,  if  sec  a:  =  |. 

11.  If  sin  a  =  0.4,  find  sin  3  a. 

12.  When  tan  a=  0.1,  find  tan  8  a. 
r--l3.  Find  sin  22|°  and  cos  22^°  from  the  functions  of  45°. 
j^l4.  Find  tan  15°  from  the  functions  of  30°. 

15.  Find  the  sin  67^°  from  the.  functions  of  22|°. 

Verify  the  following  identities : 

16.  (sin  X  ±  cos  a:)2  =  1  ±  sin  2  x. 

17.  cos^  X  —  sin*  X  =  cos  2  x. 

2  tan-  /  1  — tan^- 

18.  sina;  = j    19.  cos  a;  = 

l  +  tan2|  l  +  tan2| 

«^    o   •        .     •    o  2  sin^  X 

20.    2 sin  X  +  sin  Ax  = 


1  —  cos  X 
t'^21.    tan  a:  +  cot  a?  =  2  CSC  2  a:. 

.--V  22.   tan  (-+-)=  sec  a:  +  tana;. 

V4    2y 

23.    cot  X  —  tan  a;  =  2  cot  2  x. 


66 


PLANE   TRIGONOMETRY 


[§§  37-38 


24. 


25. 


sm  L'  X 
cot  a; 

sin  3  X 


=  1  —  cos  2  X. 
=  1  -H  2  cos  2  a;. 


26.    Sin 


w 


versa; 


27.  tan  - 


1  +  sin  X  —  cos  X 


sin  a:  2      1  -f-  sin  x  -f  cos  x 

28.  sin  4  ^  =  4  sin  ^  cos  Q  (cos^  ^  —  sin^  ^) 

=  8  cos^  6  sin  ^  —  4  cos  ^  sin  B. 

29.  cos4(9  =  8cos*^-8cos20  +  l. 

30.  When  sin 3 ^  —  sin 6 ^  =  0,  show  cos Z0=\. 

31.  sin  5  <9  =  16  sinS  ^  _  20  sin^  ^  +  5  sin  6. 

32.  cos5^  =  16cos^^  — 20cos^^  +  5cos^.     ' 

- '  33.   sin2  ^  _  sin2  ^  ^  gin  (^  j^  ^  ^^  g^n  (^  —  j5) . 

34.  cos^ A  —  cos2 ^  =  sin  (^  +  ^) sin {B  —  A). 

35.  cos2  J.  —  sin^  B  —  cos  {A  +  i?)  cos  (J.  —  B^. 

36.  sin  -  4-  cos  -  =  ±  vT+  sin  6. 


e 


0 


37.   sin  ^  —  cos  -  =  ±  Vl  —  sin  d. 
(^xplain  use  of  double  sign  in  36,  37.) 

^^-^r^'  SUM  AND   DIFFERENCE   FORMULAS 

^^  38.    Converting  to  Products.       Taking  the  results  (1),  (2), 

(3).  (4),  §  34, 

sin  (a  +  yS)  =  sin  a  cos  /8  +  cos  a  sin  /8, 
sin  (a  —  yS)  =  sin  a  cos  yS  —  cos  a  sin  yS, 
cos  (a  +  y8)  =  cos  a  cos  yS  —  sin  a  sin  /8, 
cos  (a  —  y8)  =  cos  a  cos  /3  +  sin  a  sin  yS, 

we  have  on  adding  and  subtracting  : 

sin  (a  +  /8)  +  sin  (a  —  yS)  =  2  sin  a  cos  yS, 
sin  (a  +  y8)  —  sin  (a  —  /Q)  =  2  cos  a  sin  /3, 
cos  (ce  +  yS)  +  cos  (a  —  /3)  =  2  COS  a  COS  /S, 
cosXa  +  /3)  —  COS  (a  —  /S)  =  —  2  sin  a  sin  /S.. 

Let  a  +  y3  =  X,  a  -  y8  =  F,  then 


(I) 


JT+F 


/3  = 


X-F 


§§38-39]  FUNCTIONS   OF   TWO   ANGLES  67 

Making  these  substitutions,  we  have  : 

(19)  sin .Y  +  sin  1=2 sin ^+  ^cos ^'Z  ^, 

(20)  sin  A^ -  sin  1^  =  2 cos ^+  ^ sin "^  ~  ^  , 

(21 )  cos  X  +  cos  F  =  2  cos  ^^^cos  "^  ~  ^, 

(22)  cos X -  cos  r  =  -  2  sin ^^  ^sin^~  ^ 

These  formulas  should  be  remembered  as  the  sum  of  two  sines 
equals  twice  the  sitie  of  the  half  sum^  times  the  cosine  of  the  half 
difference^  and  so  on  for  the  remaining  formulas.  Formulas 
(19)-(22)  are  the  so-called  Sum  and  Difference  Formulas; 
when  read  forward  they  convert  a  sum  or  difference  of  two 
sines  or  cosines  into  a  product ;  when  read  backwards  they  con- 
vert a  product  of  two  sines  or  cosines  into  a  sum  or  difference. 

39.  Converting  to  Sum  or  Difference.  The  formulas  (19),  (20), 
(21),  (22)  should  be  recognized  when  read  conversely.  Writ- 
ing formulas  (I),  §  38,  conversely,  and  replacing  a  by  ^,  y8  by  B^ 
we  have  : 

(19')  2  sin  ^  cos  -B  =  sin  {A  +  B)+  sin  {A  -  B), 

(20')  2 cos  A^mB  =  sin  (A  +  B)  -  sin  (A  -  J5), 

(21')  2  cos  ^  cos  iJ  =  cos  (A  +  B)  +  cos  (A  -  B), 

(22')  2 sin  A^mB  =  cos  {A-B)  -  cos  {A  +  B\f 

a  set  of  important  relations  which  should  be  recognized. 

EXERCISES 

Read  the  following  exercises,  applying  the  sum  and  difference 
formulas,  reducing  answers : 

1.  sin  10° -I- sin  40°  =  8.  sin  (- 10°) -f  sin  40°  = 

2.  sin  80° -f- sin  30°=  9^  cos  80°  -  cos  (- 20°)  = 

3.  cos  60° -f  cos  40°  = 

4.  sin  70°  -  sin  40°  =  10.  cos|  +  cos|  = 

5.  cos  28°  +  cos  42°  = 

6.  cos  28° -cos  42°=  ^^^  sin  4^  + sin  3^  = 

7.  sin  35°  +  cos  25°  =  12.  sin  3  ^  -  sin  (9  = 


58  PLANE   TRIGONOMETRY  [§39 

13.  COS  (^n -\- 1}  6  —  COS  {n  —  1)  6  =s 

14.  sin  (ji  +  1)  -  —  sin  (n  —  1)  -  == 

Express  as  a  sum  or  difference : 
-^  15.    2  sin  40°  cos  20°  =  18.    2  cos  60°  cos  10°  = 

jl.  16.    2  cos  50°  sin  40°  =  19.  '  2  sin  ^  cos  ^  = 

1^  17.    2  sin  20°  sin  40°  =  20.    2  sin  (n^)  cos  (w  -  1)  ^  = 

-^rove  the  following  identities:  -^-^ 

'  ^  sin  2x  +  sin  2  y  _  tan  {x  +  y) 

sin  1x  —  sin  2  ?/      tan  {x  —  ^) 

,^         .  ^^     cos  3  a:  —  cos  5  a;      , 

J\    ,  -- hw  22.    — — : — - — =tana;. 

A^    ^\X  sin  3  a;  +  sin  5  x 

''    y  ^23^    sinar  +  siny^^^^^(a;  +  y)^ 

Y^  cos  X  +  cos  t/  2 


24. 


25. 


sin  9  x  +  sin  a;       ,       r 

■ =  tan  6  X. 

cos  y  x  +  cos  a; 


tan 


^  +  .y 


sin  a;  +  sin  i/  _  2 


26. 


>/Jf-^   't''   ' 


Sin  a;  —  sin  ^/      ,       a;  —  y 
2 

cos  X  —  cos  y  _  _  I.      x  +  y  ,      X  —  y 

cos  a;  +  cos  y  2  2 

27.    sinfa^H — j  +  sin  [  a;  —  —  j  =  sin  a;. 
1^  v^28.    sin  (30°  +  A)  +  sin  (30°  -A)  =  cos  A. 

29.  COS  (^  +  A  j  +  COS  r^  —  A\  =  COS  ^. 

sin  Cw  —  2)  a:  +  sin  (nx^  , 

30.     ^^ ^ — ^^^ — -  =  cot  X. 

COS  (w  —  2)  a:  —  cos  (wa;) 


31.    2  sin  (  a*  +  -  )  sin  f  a;  —  -  j  =  sin^  a;  —  cos^  x. 

^  32.    cos  X  +  COS  3  a;-}-  cos  5  a;  -f  cos  7  a;  =  4  cos  x  cos  2  a;  cos  4  a:. 
33.    sin  a;  +  sin  3  a;  -f-  sin  5  a;  -f-  sin  7  a;  =  4  cos  x  cos  2  a;  sin  4  x. 


§39] 


FUNCTIONS  OF   TWO   ANGLES 


59 


34.  sin  6  +  sin  cf)  -\-  sin  (^  +  0)  =  4  cos  -  cos  ^  sin     "]"  ^ 

35.  sin  6  -f  sin  ^  —  sin  (^  +  ^)  =  4  sin  -  sin  ^  sin  -^t_2. 


36. 


sin  ^  4-  sin  ^  +  sin  i/r  —  sin  (^  +  ^  +  i/r) 
cos  6  ■+■  cos  ^  +  cos  i/r  +  COS  (^6  -\- sjr  +  (f>) 

=  tan  ^+^  tan  ^±:^  tan  ±±^. 
2  2  2 


When  A  +  B+  C=  180°,  prove  : 

J  y  ABC 

37.  sin  ^  +  sin  B  +  sin  C  =  4  cos  —  cos  —  cos — • 

2  2         2 

38.  sin  A  +  sin  B  —  sin  C  =  4  sin  —  sin  —-  cos  —  • 

^  Ja  Zk 

ABC 

39.  COS  A  +  cos  B  +  COS  (7=1  +  4  sin  —  sin  —  sin  —  • 

2         2         2 

A  ABC 

J   40,   COS  A  +  COS  B  —  cos  C  =  —  1  +  4  cos  —  cos  —  sin  —  • 

AAA 

41.  tan  A  +  tan  B  +  tan  (7  =  tan  A  tan  jB  tan  (7. 

42.  Show  ^"^  (^  +  ^^  =    cot  a:  +  cot  y  _ 

cos  (x  —  y)      1  +  cot  a:  cot  y 

43.  Show  ^^^  ^  +  ^'^"  -^  ^  ^^"  ^"^  +  '^^ 

tan  X  —  tan  ?/      sin  {x  —  y) 


44.    Show 


tan  b  X  —  tan  3  x         tan  3  a:  —  tan  a; 


1  +  tan  5  a;  tan  3  a;      1  +  tan  3  x  tan  x 

45.  Prove  that  in  a  given  circle  the  area  of  a  regular  inscribed 
pentagon  is  to  the  area  of  a  regular  inscribed  decagon  as 
cos  36°  is  to  1. 


\ 


^^-^^r' 


CHAPTER   VI 
LOGARITHMS 

Calculations  involving  multiplication,  division,  raising  to 
powers,  and  extracting  roots  may  become  quite  laborious  when 
performed  in  the  ordinary  way.  To  abbreviate  the  work  of 
numerical  calculation  certain  tables  known  as  Logarithmic  Tables 
of  Numbers  and  of  the  Trigonometric  Functions  have  been 
prepared.     The  use  of  such  tables  will  now  be  explained. 

40.  The  Index  Laws.  Let  it  be  required  to  find  an  approxi- 
mate value  of  the  fraction 

J,  ^25  X  78.6  X  tan  65° 
~       658.4x4.75 

Since  10^  =  10  and  10^  =  100,  we  may  approximate  a  number 
between  1  and  2  which,  when  used  as  an  exponent  of  10,  will 
produce  25.  This  exponent  is  approximately  1.39794.  The 
number  10  may  be  affected  by  an  exponent  which  will  produce 
any  one  of  the  numbers  in  the  fraction  F. 

We  write  the  above  fraction  in  exponential  form  thus : 
25  X  78. 6  X  tan  65°      1 0^-^'^  x  10189542  ^  iQcssiss 


F  = 


658.4  X  4.75  loa-smg  ^  ioo-67669 

1 01-39'94+1.89542+0.33133 
~~  ;[Q2.81849+0.67669 

203-62469 


_  203-62469-3.49518 


203-49518 
_  200-12951^ 

This  final  result  will  evidently  be  a  number  smaller  than  10 
since  the  exponent  is  less  than  unity ;  its  approximate  value  is 

^  =  1.3474. 

Thus,  the  operations  of  multiplication  have  been  replaced  by 
addition,  and  the  operation  of  division  has  been  performed  by 
subtraction. 

60 


§§40-41]  LOGARITHMS  '  61 

The  exponents  of  10  in  this  illustration  obey  the  iyidex  laws 
of  elementary  algebra,  viz.  (1)  the  exponent  of  a  product 
equals  the  sum  of  the  exponents,  (2)  the  exponent  of  a 
quotient  equals  the  difference  of  the  exponents  (the  exponent 
of  the  denominator  subtracted  from  that  of  the  numerator). 
To  these  two  laws  we  attach  a  third,  illustrated  by 

252=  (^101-39794)2  =  102-79588  =  625. 

These  three  index  laws  are  expressed  by  formula  thus : 

(1)  a""  xav  =  a^+v, 

(2)  a^'^ay  =  a*-y, 

(3)  Ca*)»»  =  a»»*. 

41.   Definition  of  Logarithms.* 

(1)  Common  Logarithms.  The  logarithm  of  a  number  N  to 
the  base  10  is  the  exponent  by  which  10  must  be  affected  to  produce 
the  7iumber  N.  Logarithms  constructed  upon  10  as  a  base  are 
called  Common  logarithms  (Briggian  logarithms). 

As  illustrations  of  logarithmic  notation, 

25  =  W"^'^',  log  25  =  1.39794, 

^8.6  =  lQ^-^\         log  78.6  =  1.89542, 

tan  65°  =  10"  ^^iss,     log  tan  65°  =  0. 33133. 

(2)  Logarithms  to  Any  Base.     If  we  write 

we  define  x  =  the  logarithm  of  N  to  base  a, 

or  more  briefly  lo^a  X  =  x. 

The  logarithm  of  a  number  N  to  base  a  is  the  exponent  by  which 
a  must  be  affected  to  produce  N. 

♦Logarithm  =  ratio  number.  Logarithms  were  invented  by  John  Napier  (1550- 
1617),  a  native  of  Scotland.  To  his  system  of  logarithms,  published  1614,  he  applied 
the  name  artificial  numbers.  Napier's  system  of  logarithms  was  not  constructed 
upon  10  as  a  base.  Henry  Briggs  (1556-1631),  professor  in  Gresham  College,  London, 
first  used  10  as  a  base  (1617),  and  thus  constructed  a  system  adapted  to  the  ordinary 
decimal  notation.  The  tables  first  published  by  Briggs  were  constructed  to  14  decimal 
places.  Tables  in  ordinary  use  are  constructed  to  seven,  six,  five  or  even  four 
decimals.  Greater  degrees  of  accuracy  will  be  obtained  by  using  tables  with  greater 
numbers  of  decimals.  The  ordinary  logarithms  of  the  trigonometric  functions  were 
first  constructed  by  Gunter,  1020. 


62  PLANE   TRIGONOMETRY  [§§41-43 

Examples,  i.  If  the  base  is  5,  what  is  logg  25,  logg  125, 
log5(0.2),  log, (0.04)?  Ans.    2,  3,  -1,-2. 

2.  With  8  as  base,  what  is  logg64:,  log8(|),  logg4,  logglG, 
logs  32,  logs  128?  ^n8.2,  -l,|,|,|,f 

3.  With  10  as  base,  log  2  =  0.30103,  log  3  =  0.47712, 
log  5  =  0.69897.  Find  log  4,  log  6,  log  9,  log  12,  log  15,  log  18, 
log  20. 

^Ans.  0.60206,  0.77815,  0.95424,  1.07918,  1.17609,  1.25527, 
1.30103. 

4.  From  example  3  above  find  log(|),  log(\^-),  log  1.5, 
log  1.2.  Ans.    0.35218,0.87506,0.17609,0.07918. 

5.  With  16  as  base,  what  numbers  correspond  to  the  follow- 
ing logarithms  :  2,  1  |,  ^,  —  i    —  f  ?     Ans.    256,  4,  64,  2,  ^,  |. 

42.  Systems  of  Logarithms.  Any  number  (excluding  0  and 
1)  may  be  taken  as  a  base  for  a  logarithmic  system.  Two 
bases  are  in  use,  10  and  e,  where  e  is  defined  by 

*  =  »+hiT2+TT|T3  +  r2^+      =^-'"^"S3 

Logarithms  constructed  with  10  as  base  are  called  common 
logarithms  or  Briggian  logarithms.  This  system  is  adapted  to 
all  ordinary  numerical  calculations. 

Logarithms  constructed  with  e  as  base  are  called  natural 
logarithms^  or  Napierian  logarithms.,  or  hyperbolic  logarithms. 
This  system  is  used  in  the  theoretical  work  of  higher  mathe- 
matics. Tables  of  logarithms  constructed  to  the  base  10  will 
be  found  in  the  Tables  compiled  at  the  end  of  this  book. 

43.  Laws  Governing  the  Use  of  Logarithms.  The  interpreta- 
tion of  the  Index  Laws  into  logarithmic  form  furnishes  the 
necessary  rules  for  performing  multiplication,  division,  raising 
to  powers,  and  extracting  roots  by  means  of  logarithms. 

I.  Law  of  Products.  The  logarithm  of  a  product  equals 
the  sum  of  the  logarithms  of  the  factors. 

For,  let        .  a^  =  IV,     a^  =  M, 

then  '  x=log„]Sr,     i/  =  log„M. 

Also,  N  X  M=  a""  xa^  =  a^+^, 

log„ (Nx  M)  =  iK  +  y=  log„ i>r+  log„  Mf 

which  establishes  the  law. 


§§  43-44]  LOGARITHMS  63 

Example.     logio(67  x  126)  =  log^^  67  +  \og^^  126 

=  1.82607  +  2.10087  =  3.92644. 

II.  Law  of  Quotients.  The  logarithm  of  a  quotient  equals 
the  logarithm  of  the  numerator  diminished  by  the  logarithm  of  the 
denominator. 

For,  let  «-^  =  iV,     a^  =  3f, 

then  N-^  M=  a^ -i- a^  =  a'-y. 

In  logarithmic  form 

***^«  (^)  "  ^  ~  ^  "  **^«  '^  ~  ^®^«  ^^' 
Example,     log^^  (184  -^  1.56)  =  log^^  184  -  log^o  1.56 

=  2.26482-0.19812  =  2.07170. 

III.  Law  of  Powers  (Roots).  The  logarithm  of  N" 
equals  n  times  the  logarithm  of  N.  This  law  is  true  for  any 
value  of  the  exponent  n,  whether  positive  or  negative,  integer 
or  fraction. 

Let  a^'^N,   x  =  \og„N', 

then,  by  the  third  index  law, 

(a^)"  =  a'""  =  iV^". 

In  logarithmic  notation. 

Examples.  log  507-^  =  8  log  507, 

log(148)i  =  Hogl48, 
log  (576)*  =  I  log  576. 

44.  Characteristic  and  Mantissa.  From  the  table 
10*  =10000 
103  ^  1000 
102  =  100 
101  =  10 
100    ^  1 

10-1=        .1 
10-2=       .01 


64  PLANE   TRIGONOMETRY  [§44 

it  is  obvious  that  the  logarithm  of  an  integral  power  of  10  is  an 
integer,  either  positive  or  negative.  The  logarithms  of  num- 
bers between  1  and  10  are  evidently  between  0  and  1,  loga- 
rithms of  numbers  between  10  and  100  are  between  1  and  2, 
and  so  on.  For  example,  log  8  =  0.90309,  log  80  =  1.90309, 
log  800  =  2.90309,  log  8000  =  3.90309. 

The  integral  part  of  a  logarithm  is  called  the  characteristic ; 
the  decimal  part  of  the  logarithm  is  called  the  mantissa. 

I.  Law  of  the  Characteristic.  From  the  above  exam- 
ple, it  is  seen  that  log  8  has  a  characteristic  0,  log  80  has  1, 
log  800  has  2,  and  so  on.  From  this  illustration,  we  see  tliat 
the  characteristic  of  a  logarithm  of  a  whole  number  is  one  less  than 
the  number  of  digits  in  the  number. 

Dividing  a  number  by  10  reduces  the  characteristic  by  1 ; 
dividing  by  100  reduces  its  characteristic  by  2,  and  so  on. 
Hence,  for  a  pure  decimal,  as  .00567,  the  characteristic  is  nega- 
tive and  equal  to  the  number  of  places  which  the  first  significant 
figure  occupies  to  the  right  of  the  decimal  point.     Thus, 

log  0.00567  = -3 +  .75358, 
log  0.0258  =-2 +  .41162. 

In  such  cases,  the  characteristic  is  negative,  while  the  mantissa 
is  positive.  The  usual  notation  is  to  place  a  negative  sign 
(  — )  over  the  characteristic,  leaving  the  form  for  the  above 
illustrations  thus : 

log  0.00567  =  3.75358, 

log  0.0258  =  2.41162. 

II.  Law  of  the  Mantissa.  The  mantissa  is  the  same  for 
any  given  sequence  of  digits^  whatever  may  be  the  position  of  the 
decimal  point. 

For  example,  log  4896  =  3.68984,  log  48.96  =  1.68984, 
log  4.896  =  0.68984,  log  0.04896  =  2.68984. 

Logarithmic  tables  as  usually  constructed  show  only  the 
mantissa.  The  characteristic  is  to  be  supplied  according  to 
Rule  I. 


§  45]  LOGARITHMS  65 

45.  Use  of  Tables.  Tabulation  of  Logarithmic  Work.  (1)  Ty 
find  the  logarithm  of  a  number.  Let  us  find  the  logarithm  of  a 
number  of  five  figures. 

Take  for  example  log  58,769. 

(a)  The  characteristic  is  4. 

(b)  Look  in  coluinu  marked  N,  of  the  Tables,  for  the  first  three  digits 
587 ;  follow  the  horizontal  row  opposite  587  to  the  right  to  column  6  at  top, 
and  we  find  the  mantissa  .76908. 

(c)  The  correction  for  the  last  figure  9  is  approximately  .9  of  the  differ- 
ence between  the  mantissa  for  5876  and  that  for  5877,  i.e.  .9x8  =  7.2. 
Hence,  add  7  to  the  last  figure  of  the  mantissa  already  found. 

log  58769  =  4.76915. 

(d)  In  the  margin,  the  tabular  differences  between  any  two  successive 
mantissas  are  indicated ;  on  this  page  these  differences  are  7  and  8.  The 
corrections  for  1,  2,  3,  •••,  9  are  indicated  in  the  columns  under  the  8  and  7, 
respectively. 

(2)  To  find  the  number  corresponding  to  a  logarithm.  A  num- 
ber whose  logarithm  is  given  is  found  by  reversing  the  above 
process. 

Required  the  number  corresponding  to  the  logarithm  2.57682. 
log  N  =  2.57682. 

(a)  In  the  logarithmic  tables  find  the  mantLssa  next  smaller  than  .57682. 
This  is  .57680,  leaving  a  difference  2. 

(b)  The  digits  in  column  N  and  at  top  of  column  in  which  .57680  is 
found  are  3774. 

(c)  The  difference  between  .57680  and  the  next  tabular  number  is  12. 

(d)  In  proportional  parts  column,  under  12,  the  nearest  number  to  pro- 
portional pai't  2  is  2. 

(e)  Hence,  N  —  377.42,  the  decimal  point  falling  between  7  and  4,  since 
the  characteristic  is  2. 

(3)  To  find  the  logarithm  of  trigonometric  functions. 

(1)  To  illustrate,  let  us  find  the  log  sin  37^  48'  1.5". 

(o)  Sines  and  cosines  are  less  than  unity ;  hence,  the  characteristic  in 
each  case  is  negative.  To  avoid  negative  characteristics  in  the  tables,  10  is 
added  and  subtracted. 

(6)  log  sin  37°  48'  =  9.78739  -  10.     Correction  for  15"  =  4.     Hence, 
log  sin  .37"  48'  15"  =  9.78743  -  10. 

(c)  The  tabular  difference  is  found  in  column  marked  d,  and  proportional 
corrections  for  6",  7",  8".  9",  10",  20",  30",  40",  50"  are  indicated  in  the 
margin  under  proportional  parts. 


PLANE   TRIGONOMETRY 


[§45 


(2)  Find  log  cos  49^  21'  30". 

(a)  The  log  cos  49°  21'  is  found  by  looking  at  bottom  of  page  69,  reading 
up  the  column  marked  log  cos  and  taking  the  minutes,  21',  from  the  column 
to  right  of  the  page. 

log  cos  49°  21'  =  9.81387  -  10. 

(6)  Since  the  cosine  decreases  as  the  angle  increases,  the  correction  for 
30"  must  be  subtracted. 

Correction  30"  =  7.5  (count  as  8)  ;  see  Prop.  Pts.  under  15. 
(c)  Hence,  log  cos  49°  21'  30"  =  9.81379  -  10. 

The  use  of  the  tables  in  finding  logarithms  of  tangents  and 
cotangents  is  similar  to  that  explained  for  sines  and  cosines. 


W 


TABULATION  OF  WORK 


In  the  work  of  computation  by  means  of  logarithms  it  is 
'very  important  that  the  computor  should  economize  in  both 
time  and  labor.  To  accomplish  this  purpose  the  beginner 
should  (1)  make  himself  familiar  with  the  mechanical  con- 
struction of  the  logarithmic  tables,  especially  the  de^dces  of 
marginal  corrections,  and  (2)  he  should  adopt  and  systemati- 
cally carry  out  some  convenient  plan  of  tabulation  of  his  work. 

A  plan  of  tabulation  is  suggested  in  the  following  examples. 

Examples,     l.    Find  by  means  of  logarithms  the  value  of 
a  X  h  X  tan  a 


F=  , 

c  X  a 

and  a  =  65°. 


-,  where  a  =  25,  6=78.6,  c  =  658.4,  £?  =  4.75, 


Let  the  numerator  be  called  N^  the  denominator  D. 
we  may  tabulate  as  follows : 

a  X  5  X  tan  a  _N 


Then 


F  = 


c  X  d 


Data  and  Mesultg 


Logarithms 


a 

25 

h 

78.6 

a 

65° 

c 

658.4 

d 

4.75 

F 

1.3474 

log  a 

log  ft 

log  tan  a 


loge 
log  d 

log^ 


1.39794] 
1.89542 1 add 
0.331331 


3.62469 

2.81849] 

0.67669] 

3.49518 

0.12951 


add 


J2 


I  45-46] 
2.    FindJP  = 


LOGARITHMS 


67 


a  sin  a 


h  tan  /8 
and  /3  =  70°  41'  10". 

Data  and  Results 


,  where  a  =  4694.5,  6  =  37.6,  a=67°20'  15" 
Logarithms 


a 

4694.5 

a 

67°  20'  15" 

h 

37.6 

fi 

70°  41'  10" 

F 

40.377 

log  a 
log  sin  a 


log  a  sin  « 


3.67154 

5  correction 
9.96509-1  0 

1  correction 


log  h 
log  tan  )8 


log  h  tan  )8 


losr^ 


13.63669-10 


1.57519 
0.45529 

6.7  correction 


2.03055 


1.60614 


46.    Conversion  of  Common  to  Napierian  Logarithms.     Loga-  .  w 

rithms  of  one  system  may  easily  be  converted  into  logarithms"/  ^^^"^ 
of  another  system.     Let  us  consider  the  systems  to  base  10 
and  e.     Let 

10y  =  N,  then  y  =  log^QN. 

Take  logarithm  to  base  e  of  both  members  of  10^  =  N. 

y  log,  10  =  log,  iV, 
or  logio  N   loge  10  =  loge  ^• 

Now,  log,  10  =2.302585;  hence, 

(1)  loge^V=  2.302585  x  log^o^. 


(2) 


logio^"  = 


2.302685 

EXERCISES   IN   USE   OF   LOGARITHMS 


X  loge  ^  =  0.434294  x  logc  ^' 


By  means  of  logarithms  find  the  value  of  each  of  the  fol 
lowing : 

78.54  X  9.6752 


1.  Q  = 

2.  Q  = 


8.269 
(104.6)^  X  0.2536 
(5.87)^ 


Ans.    91.896. 
An%.    0.49319. 


4/^M^^^^^- 


68  PLANE   TRIGONOMETRY  [§46 

3     (._  (1848)^x10-^  ^ 

^-    ^"         0.7854  ^'*''    ^•^^^^• 

^     ^      1496  X  tan40°rr'  .         ..^^  . 

cos  10  zV 

5.  ^  =  32.6  X  5.87  X  sin  56°  24'  15".  J.ws.    159.396. 

6.  ^=7rr2,  7r=  3.1416,  r=  3.6.  Ans,    40.715. 

7.  The  area  of  a  triangle  is  given  by  A  =  |  ab  sin  O,  where 
2,  5  are  sides  and  0  is  the  included  angle.  Find  A,  when 
a  =  46.7,  J  =  6.91,  (7  =  38°  24' 10".  Ans.    100.23. 

8.  The  volume  of  a  sphere  is  given  by  V  =  ^  "rrr^ ;  find  V 
when  7r  =  3.1416,  r  =  8.65  cm.  Ans.    2711.1  cu.  cm. 

9.  Mass  =  volume  x  density.  Find  the  radius  of  a  sphere 
if  its  mass  be  4.38  x  10^  grams  and  its  density  2.3. 

Ans.    35.69  cm. 

10.  The  area  of  a  segment  of  a  circle  is  given  by 
S  =  ^7^{6  —  sin  ^),  where  the  radius  is  r,  and  0  is  the  angle 
subtended  by  the  segment  at  the  centre  of  the  circle.  Find  S, 
when  r  =  4.6  ft.,  6  =  126°  30'.  Ans.    14.85  sq.  ft. 

11.  The  volume  of  a  right  circular  cone  is  given  by  V=  |  TrU^h, 
where  R  is  the  radius  of  the  base  and  h  is  the  altitude.  Find 
V,  when  R  =  1.5876  m.,  and  h  =  7.675  m. 

Ans.   20.257  cu.  m. 

12.  If  the  volume  of  a  cone  V=  987.6  cu.  cm.,  and  its  height 
h  =  9.416  cm.,  find  the  radius  of  the  base.  Ans.    10  cm. 

13.  The  time  of  vibration  of  a  pendulum  is  given  by  t  =  ttV-  , 

where  I  =  length  of  the  pendulum,  g  =  the  acceleration.     Find 
the  length  ?  of  a  pendulum  vibrating  seconds  if  </  =  980.19  cm. 

Ans.    99.314  cm. 

14.  What  value  must  g  have  in  order  that  a  pendulum  1  m. 
long  shall  vibrate  seconds?  Ans.    986.96  cm. 

15.  Find  the  length  of  a  pendulum  I  which  makes  80  vibra- 
tions per  minute  ;  g  =  32.16  ft.  Ans.    1.8329  ft. 

16.  Find  the  time  of  vibration  of  a  pendulum  if  ?=  8.04  ft., 
^  =  32.16-ft.  Ans.    1.57  sec. 


CHAPTER   VII 


SOLUTIONS    OF    TRIANGLES    IN    GENERAL 

We  shall  now  develop  a  number  of  theorems  which  are  used 
in  the  solution  of  any  plane  triangle.  The  demonstrations  of 
these  theorems  should  be  thoroughly  mastered. 

B 
47.    The  Theorem  of  Sines. 

(1)  First  demonstration. 

Uniform  lettering  of 
a  triangle  as  indicated 
in  the  drawings  will  be 
found  convenient.  Let 
the  angles  be  indicated 
by  capitals  A^  B,  C, 
and  let  the  sides  oppo- 
site be  called  a,  h,  e, 
respectively.     Draw  a  perpendicular  p  from  the  vertex  upon  the 


i'ig.  35. 


opposite  side  5,  Fig.  34,  or  b  produced,  F'ig.  35.     Then,  from 
the  right  triangles  AXB  and  CXB^  we  have 


inA^P, 


sin 


sin 


\ 


a 


[§i7 


(I) 


70  PLANE   TRIGONOMETRY 

Dividing  sin  A  by  sin  C, 

sin  A  _a 

sin  C     c 

In  Fig.  35,  we  have 

sin(180°-(7)=sinC=^. 


Drawing  a  perpendicular  from  angle  A  to  side  a,  we  would 
have,  similarly, 

sin  C     c 


Dividing  (I)  by  (II), 


sin  A  _a 
sin  B      h 


(III) 


These  results  may  be  written  in  the  symmetrical  form 

sin  A     sin  B     gin  C 


Theorem.  In  any  triangle  the  sines  of  the  angles  are  'propor- 
tional to  the  opposite 
sides. 

(2)  Second  demon- 
stration. The  theorem 
of  sines  may  also  be 
proved  by  circumscrib- 
ing a  circle  about  the 
given  triangle,  Fig.  36. 
Draw  the  diameter  AD 
=  2  R,  and  the  line  BD 
forming  the  right  tri- 
angle ABB.  Then,  the 
angle  ABB  =  angle  0. 

sin  ABB  =  sin  (7  =  — -  • 
IB 


Fig, 

.36.                                       Sin^ 

Similarly, 

1         a        •     rt        b 

Hence, 

sin  A     sin  ^     sin  C       1 
a            b           c         2iJ' 

where  R  is  the  radius  of  the  circumscribing  circle. 


§47] 


SOLUTIONS   OF   TRIANGLES   IN   GENERAL 


71 


It  should  be  observed  that  tlie  theorem  of  sines  may  be  em- 
ployed in  the  solution  of  a  triangle  when  two  angles  and  a  side 
are  given,  or  when  two  sides  and  an  angle  opposite  one  of  them 
are  given. 


APPLICATIONS   OF    THEOREM   OF   SINES 
Case  I.      G-iven  a  side  and  two  angles. 

Example.     Given  a  =  148.3,  A  =  37°  24',  C=  76°  48'  80",  to 
find  h  and  c. 

Solution.    (1)  Since 
A  +B  +  C  =  180°, 
ii=180°-  (A  +  C) 
=  65° 47' 30". 
(2)    Make  an  approximate  con- 
struction of   the  triangle.     (3) 
Select  formulas,  and  tabulate  the 

calculations. 

,      a  X  sin  B 


Given 


sin  A 

a  X  sin  C 

sin  A 

Data  and  Results 

a 

148.3 

A 

37°  24' 

C 

76°  48'  30" 

B 

65°  47'  30" 

Logarithms 


logo 
log  sin  A 


222.695 
237.72 


log(^) 
\sin  At 

log  sin  j5 

log  sin  C 


log  6 
logc 


2.17114 
9.78346 


2.38768 

9.96003 
9.98839 


2.34771 
2.37607 


Case  II.      Given  two  sides  and  an  angle  opposite  one   of  the 
given  sides.     This  problem  may  have  (1)  one  solution,  (2)  two 


Fis.  38. 


79 


PLANE   TRIGONOMETRY 


[§47 


solutions,  (3)  or  no  solution.  If  the  dimensions  given  be  a,  c, 
and  A,  where  A  is  acute,  one  solution  will  result  if  a  >  c,  or  if 
csin^  =  a.  Fig.  (1),  (2);  hco  solutions  (triangles  ABC,  ABC) 
will  occur  when  a<c  and  a  >  c  sin  A,  Fig.  (-3).  li  a<e  sin  A, 
no  solution  will  exist.  Fig.  (4).  If  A  be  obtuse,  one  solution  will 
exist  provided  a  >  c,  otherwise  the  construction  is  impossible. 

Example.     Given  a  =  556,  b  =  678.4,   A  =  31°  10' 30",  to 
find  B,  a  c. 

In  this  example  the  angle  A  is  acute  and  the  side  a  <  6,  hence 

-B. 


two  solutions  will  be  found.     In  one  solution  the  angle  B^  will 
be  acute,  in  the  other  solution  B^  =  180°  —  By 

Solution.     (1)  Make  an  approximate  construction  of  the  data. 

(2)  Formulas  and  tabulation  of  the  calculations. 

.     7>      b  X  sin  A     ^      i  q^o      y  a    ,   d\  ax  siii  C 

smB  =  -^ ,   C  =  180° —(A  +  B),  c  =  — : 

a  sm  A 


Data  and  Results 


Logarithms 


Given 


a 

h 

A 

556  . 
678.4 
31° 10' 30" 

B, 

39° 10' 12" 
140°  49' 48" 

109°  39'  18" 
7°  59'  42" 

^2 

1011.5 
149.39 

logo 
log  sin  A 
log  6 

2.74507 
9.71404 
2.83149 

log  sin  B 

9.80046 

log  sin  Ci 
log  sin  C^ 

9.97393 
9.14329 

logci 
logcg 

3.00496 
2.17432 

§§47-48]      SOLUTIONS   OF   TRIANGLES   IN   GENERAL  73 

V  EXERCISES 

'  ^  1.  Given  a  =  140.6,  A  =  48°  30'  10",  B  =  76°  24';  find  6,  e. 

^         /^2.  Given  b  =  3875.4,  A  =  97°  24',  B  =  40°  27'  15" ;  find  a,  c. 

{  ^     -3.  Given  a  =  148.6,  6  =  121.78,  A  =  69°  20'  10";  find  B,  O^ 

>4.  Given  a  =  2311,  b  =  1600.7,  5  =  34°  42'  29";  find  A,  C,  c. 

5.  Given  a  =  1906,  b  =  224.8,  A  =  61°  24'  18";  find  B,  C,  c. 

Given  b  =  1009,  e=  796.4,  (7=  85°;  find  B,  A,  a. 

7.  Given  A  =  67°  54',  B  =  34°  52',  6  =  4356.7;  find  a,  c. 

48.    The  Theorem  of  Tangents.     From  the  theorem  of  sines 


^^e! 


M^ 


sin  A      sin  B      sin  C       1 


a  6  c         'IR' 

we  have  a=  2  R  sin  ^,     b  =  2  R  sin  5. 

Adding  and  subtracting,  and  reducing  by  §  38, 

(1)  a  +  b=2  R(ismA  +  sin  B)=-k  i?  sin^  ^  ^  cos^  ~  ^, 

(2)  a  —  6  =  2  ^(sin  A  —  sin  ^)  =  4  i2  cos  — ^^ —  sin  — 

Dividing  (1)  by  (2), 

« +  6  2 


a—  b     .      A 

tan  — 


We  may  derive  in  a  similar  manner,  or  write  by  symmetry, 

tan^^  tan^±^ 

b+c ^  2  c+a ^  2      . 

b-<^     tan^-:^'  *'"«     tan^^ 

2  2 

Theorem.  /«  any  triangle  the  sum  of  two  sides  is  to  their 
difference  as  the  tangent  of  the  half  sum  of  the  opposite  angles  is 
to  the  tangent  of  their  half  difference.  '  - 

Tlie  theorem  of  tangents  may  be  used  in  the  solution  of  a  tri- 
angle when  two  sides  and  their  included  angle  are  given.  For 
example,  let  a,  6,  C,  be  given.     Then  we  know 

A  +  B+C=1S0°. 


74 


PLANE   TRIGONOMETRY 


[§48 


Hence,         tan  — - —  =  cot  — , 


and  the  first  formula  above  becomes 

a  +  b 


cot- 


a  —  h 


tan 


A-B' 

2 


A  —  B 

This  formula  enables  us  to  find  the  unknown  tan — - — ,  and 


then  the  angle 


A-B 


Aa-  B 

The  angle  — -^ —  being  known,  we 


find  at  once  the  values  ,of  A  and  B  : 

A=^liA  +  B)  +  i<iA-B},   B=l(iA  +  B)-l(A-B). 
The  application  of  the  theorem  of  sines  now  determines  side  e. 


APPLICATIONS   OF   THEOREM  OF   TANGENTS 

Example. 

Given       6=1436.7, 
c=  1141.2, 
^=42°  14'  35"; 
find  B,  C,  a. 

Solution.  (1)  Make  an  approxi- 
mate construction. 

(2)  Select  formulas  and  tabulate 
the  calculations. 


b=  1436.7 
.  Fig.  40. 

tan  -  (ZJ  -  C)  =  * 


b  +  c        2 
Data  and  Results 


tan  ^(B  +  C),  B  +  C  =  180°  -  A,  a  =  ^  ^.^'^/^ 
"  sin  B 


Logarithms 


Given 


b 

c 

A 

1436.7 
1141.2 
42°  14'  35" 

h-c 

b  +  c 

295.5 
2577.9 

B  +  C 

68°  52'  42" 

2 
B  -  C 

16°  31'  40" 

•) 

B 
C 

85°  24'  22" 
52°  21'  2" 

a 

969.0 

log  (6  -  c) 

1      4-      B+C 
log  tan 


log  (ft  +  c) 


log  tan 


B-C 


log  6 
log  sin  A 
log  sin  B 
log  a 


2.47056 
0.41308 
3.41126 


9.47238 


3.15737 
9.82755 
9.99860 
2.98632 


48-49]      SOLUTIONS   OF   TRIANGLES   IN   GENERAL 


75 


V 


EXERCISES 

Solve  for  the  unknown  parts  of  the  following  triangles : 

1.  a=  281,  c  =  153,  B  =  34°  42'  29". 

2.  h  =  296,  c  =  178,  A  =  78°  21'  40". 
^3.   h  =  199.37,  c  =  642.75,  A  =  130°  9'  24". 

4.  a  =  101.47,  ^  =  9936.7,  <7=  47°  48' 12". 

;^5.  h  =  1134.7,  c  =  2277.9,  A  =  19°  34'  24". 

6.  a  =  1434.2,  f>  =  9767.2,  (7=  109°  19'  36". 

7.  ?.  =  .538,  c  =  1.245,  A  =  62°  14'  40". 

8.  b  =  234.7,  c  =  185.4,  ^  =  84°  36'. 

9.  a  =  1896.9,  5  =  3463.7,  C=  124°  10  . 
10.  ^=9.876,  a  =  4.921,  ^=76°  20.4'. 


49.     The  Theorem  of  Cosines.     (1)  First  derivation. 

Draw  the  perpendicular  p  from  B  to  b,  Fig  41.     Then,  from 
the  right  triangle  BXC  we  have 

a2  =  jo2  _i_  CX^=p^  +  (b  -  AXy=p^  +  62  +  AX^  -2b-  AX. 

But  we  have  from  B 

the  right  triangle 

AXB 

p^  +  XT'  =  e2, 
and 

AX—  ex  cos  A. 

Making     these 
substitutions,    we   A 
find 


and  by  syrnmetry, 


a-  =  6^  +  c^  —  2  he  cos  A, 

62  :=  c2  +  «2  _  2  ca  cos  J?, 
c^  =  a-  +  ft"'^  —  2  «6  cos  C 


If  the  point  X  should  fall  upon  the  base  produced,  it  may 
readily  be  shown  that  the  above  results  still  hold. 


76 


PLANE   TRIGONOMETRY 


[§49 


Theorem.  In  any  triangle  the  square  of  any  side  equals  the 
8um  of  the  squares  of  the  other  two  sides  diminished  by  twice  their 
product  into  the  cosine  of  their  included  angle. 


(2)  Second  derivation. 
From  Fig.  42,  we  have 

AX=  c  X  cos  A,  CX=  a  x  cos  O, 

and  therefore, 


Similarly, 


b  =  c  X  cos  A+  a  X  cos  C. 

c  =  a  X  cos  B+  b  X  cos  A, 
a  =  b  X  cos  C  -{-  c  X  cos  B. 


(-5) 
(a) 


Multiply  these  equations,  as  indicated,  by  (—  6),  (—  <?),  (a), 
respectively,  and  add,  giving 


or 


a^—b^-~c^=  —  '2bcx  cos  A, 
a^  =  b^  +  c^—2bc  X  cos  A. 


By  similar  manipulation,  the  other  formulas  may  be  de- 
rived. 

The  theorem  of  cosines  is  not  adapted  to  logarithmic  calcula- 
tion. When  two  sides  and  the  included  angle  are  given,  this 
theorem  gives  the  third  side.  When  the  given  sides  are 
simple  numbers  such  that  their  squares  may  readily  be  known, 
the  application  of  the  cosine  formula  is  to  be  advised.  Other- 
wise, the  theorem  of  tangents  should  be  employed. 


§49-50]        SOLUTIONS  OF   TRIANGLES   IN   GENERAL  77 

EXERCISES 

Calculate  from  cosine  theorem  the  following  without  using 
logarithms : 

4  1.   h=  12,  (?  =  15,  ^  =  45°  34' ;  find  a. 

2.   c'=10,  a  =  20,  ^=37°20';  find  5. 

-*3.    rt  =  5,  6  =  7,  c=9;  find  JL. 

-*  4.   a  =  25,  6  =  40,  c  =  60 ;  find  O.  ^ 

5.  yl  =  140°,  6  =  100,  c  =  300;  find  a.    ^^■ 

6.  a  =150,  6=180,  0=97°;  find  c. 

7.  Find   the    perimeter   of    a   triangle   where    a  =  200   ft., 
6  =  300  ft.,   0=37°  40'. 

8.  Find  the  area  of  a  square  whose  perimeter  is  the  same  as 
the  triangle,  where  a  =  40,  6  =  60,  (7=  68°  10'. 

50.   The  Half-angle  Theorems.     From  the  formula 

a2  =  52  _|_  g2  _  2  6c  X  cos  A  (I) 

we  may  derive  results  suitable  for  logarithmic  calculation  of 
the  angle  A  when  the  three  sides,  a,  6,  <?,  are  given. 
Adding,  and  subtracting  2  6c,  we  find 

a2  =  J2  _^  2  6c  +  c2  _  2  6c  -  2  6c  X  cos  J. 
=  (5  +  c)2  _  2  6c(l  +  cos^) 

=  (5  +  c)2  -  2  6cC2 cos2 ^Y  §  37 

Solving  for  cos^— ,  and  factoring  the  right-hand  member, 

^^„g  A  _ih  +  c  ^r  (i){h  ^  c  -  a) 
2  4  6c 

Now,  let  a  +  6  +  c  =  2  s, 

then,  6-t-c  —  a  =  2s  —  2«; 

substitute  and  extract  the  square  root, 

A         ls(s—a) 

cos     — '  -^ 


A        /^ 

'2  =  \' 


6c 


_  a^ 


78  PLANE    TRIGOXOMETRY  [§50 

Again,  taking  (I),  subtracting  and  adding  2  Jc, 

a2  =  52  _  2  5c  +  (?2  +  2  Jc  -  2  6<?  X  cos  ^ 

=  (i  _  c)2  +  2  hc(\  -  cos^)  =  (6  -  c)2  +  2  5c(2  sin2— y     §  37 

J.  \  2/ 

d  solving  for  sin^— , 

2  4  6<?  4  66' 


Now,  take  a4-5  +  c=2  8, 

then,  »a  —  6  +  <?  =  2«— 2  6, 

a  +  6  —  c—  28  —  2c; 

substitute  and  extract  the  square  root, 

•    A      ^l(s-h)(8-c) 

«^"v=V r. ^- 


•    A  A 

Dividing  sin  —  by  cos  — ,  we  find 


tan^=J('-^)('-"). 
2       ^       «(« —  a) 

The  form  of  this  radical  may  be  changed  so  as  to  make  it  sym- 
metrical in  a,  J,  c. 


—  a)(8  —  6)(s  —  e) 


tan^=x/S>^><^-^)=^-J<^^-^)^ 
2       ^      «(« —  a)  8  —  a  ^ 

The  radical  here  employed  will  be  the  same  for  tan  — ,  tan  — • 

'A  A 

n.                 1(8  —  a)(8  —  b')(8  —  c') 
we  set  \/^^ ^ ^-^ ^  =  r, 

^  8 

+„„^         f        +^         r        4-      G        r 
tan  —  = ,    tan  —  = -,    tan  —  = 


2      8  —  a  2      s  —  b  2      s  —  e 

Then,  we  have  as  results  the  following  half-angle  formulas. 
(1)   7^e  8ine8  of  the  half-angle8  of  a  triangle. 


A 
2 

B, 
2  " 

->- 

-b){8- 

-c) 

\ 

hc 

9 

sin 

-xS^- 

-c)(s- 

-a) 

\ 

ca 

8i„C       /(8-a)(«z^. 
2      ^  ab 


§50]  SOLUTIONS   OF   TRIANGLES   IN   GENERAL  79 

(2)    The  cosines  of  the  half-ati()les  of  a  triangle. 


z      ^      be 


^1    -jii,^ 


ca 


COS 


^  ^^/g(g  -  c) 
2      ^      ab 


(3)    T7ie  tangents  of  the  half -angles  of  a  triangle. 


l(s- 

-«)(» 

-b){s- 

-c) 

/                  s 

Ks- 

■a)  (8 

-b)(8- 

-c) 

'                  s 

l(s- 

■a){s 

-b)(s- 

-c). 

r 
» 


s-b 

tan 

z      s  —  c  '*  s  s  —  c 

In  calculating  the  angles  of  a  triangle,  the  tangents  of  the 
half-angles  should  be  used,  as  the  complete  calculation  of  A,  B, 
O  may  be  performed  by  taking  only  four  logarithms  from  the 
tables,  viz.  log  s,  log  (s  —  <z),  log  (s  —  5),  log  («  —  c). 

APPLICATION  OF   THE   HALF-ANGLE   THEOREMS 
Example.     Given  a  =  65.43,  J  =58.26,  c=  49.35;  find  A, 

B,  a 


Solu/inn.     (1)  An  approximate  construction  with  the  given   data  will 
show  angle  A  > B>  C.     (2)  Select  formulas  and  tabulate  the  calculations : 

tan^  =  _r-,    tan^  =  -!^,    tan^  =  ^il-,   ^  ^  J^E^K^ZMEf) . 
2      a— a  2      s  —  b  2      s  —  c  ^  s 


80 


PLANE   TRIGONOMETRY 


[§  50-51 


Data  and  Results 


Logarithms 


Given 


a 

(55.43 

h 

58.26 

c 

49.35 

2s 

173.04 

s 

86.52 

s  —  a 

21.09 

s-h 

28.26 

s  —  c 

37.17 

A 

37°  ir  18" 

B 
o 

29°  31'  10" 

C 
2 

23°  17'  31" 

A 

74°  22'  36" 

B 

59°  2'  20" 

C 

46°  35'  2" 

log(s  -  a) 

1.32408 

log(.f  -  b) 

1.45117 

log(s  -  c) 

1.57019 

logs 

1.93712 

logr^ 

2.40832 

logr 

1.20416 

log  tan  — 

9.88008 

log  tan  — 

9.75299 

log  tan  — 

9.63397 

As  a  check, 

A  +B  +  C  =  180°. 


EXERCISES 
Find  the  angles  in  each  of  the  following  triangles : 

c=  140.76. 
c  =  35,891. 
c  =  11.675. 


1. 

2. 
3. 

4. 
5. 


«  =  98.76,  6  =  104.97, 
a  =  57,896,  6  =  49,784, 
«  =  5.769,    6  =  9.8764, 


a  =  0.0587,  6  =  0.09765,  c  =  0.1067. 
a  =  94.28,     6  =  112.68,     c  =  180.47. 


AREAS   OF   TRIANGLES 

Many  expressions  may  be  obtained  for  the  area  of  a  triangle. 
Some  of  these  will  be  enumerated. 

51.    Area  in  Terms  of  Sides  and  Angles.      Let  the  area  be 
denoted  by  K,  and  let  the  triangle  be  lettered  as  shown  in  the 

drawing.       Then,    from 
plane  geometry, 

(1)  K=  ^hxh. 

Making  use  of  trigo- 
nometric relations 
h=  0  sin  A, 

(2)  K=^hc  sin  A. 


§  51-52] 


SOLUTIONS  OF   TRIANGLES  IN   GENERAL 


81 


Theorem.      The  area  of  any  triangle  equals  one  half  the  prod- 
uct of  two  sides  by  the  sine  of  the  included  angle. 


T^     lex  sin  B  ^.     .      .. 

z     sin  C 
_  1     <?■  sill  A  X  sin  B 
~2  sin  (7 

sm  A=  2  sill  —  cos  — , 

9  9' 


(3) 

Since 

and  .-..^       ,  j_^  ,    __.  ^ 

we  may  express  IC  directly  in  terms  of  the  three  sides 


sm 


^V 


(s  —  b)(s  —  c)  A      ^s(8—a) 

^  ^^         '     cos  —  =  \/  — ^^^ ■ 

^        be 


§47 

§37 

§50 


K=-bc  sin  A  =  bc  sin  —  cos  — 


=  ^,^(^-M(.-c)  ^ ^«i^-a} 


where 


=  Vs(«  —  a)(«  —  5)(s  —  c), 

2s  =  a4-^  +  (?. 


bo 


52.    Area   in   Terms  of  **.     The  area  of  a  triangle  may  be 

B 


Tm.  45. 


expressed  in  terms  of  the  radius  of  the  inscribed  circle.     Let 
a  circle  be  inscribed  in  the  triangle  ABC.     See  Fig.  45. 


Then, 


Adding,  we  have 


AAOO=lbr, 
ABOA  =  lcr, 
A  COB  =  I  ar. 

A  ABC=  K=  l(a  +  b-\-c)r  =  sr, 
s=lCa  +  b  +  c). 


82  PLANE   TRIGONOMETRY  [§  53 

53.    Expressions  for  the  Area  of  a  Triangle. 
1.   K  =  1  be  X  sin  A  =1  ca  X  Hin  B  =  1  ab  X  sin  C 


\  C-  sin  A  sin  B 


2.    K  = 


sinC 


3.  X  =  v's(«-a)(«-&)(s-c),  2s  =  a  +  6  +  c. 

4.  X  =  s  X  r,  r  =  radios  of  inscribed  circle. 

EXERCISES 
Solve  the  following  triangles  for  the  unknown  parts : 
>i.   a  =  7950,   ^  =  79°  59',   ^  =  44°  41'. 

2.  a  =  80.86,   ^  =  19°  29',   B=SS°V. 

3.  a  =  62.65,   6  =  89.81,    (7=  55°  5'. 

4.  a  =2071,    6=1887,    (7=  55°  12' 3". 

5.  a  =  0.2034,    5  =  0.112.3,    (7=  72°  15' 19". 

6.  a  =  48.5,    6=84,   ^=21°  31'. 

7.  a  =838.56,   6  =  841.53,   ^=68°  10' 24". 

8.  a  =  49,    6  =  45,    j5=17°41'9". 

,r  9.  6  =  117.4,   c  =  726.3,   ^  =  80°  10'. 

^10.  c=  1047.51,   a  =  943.27,   ^=63°  17' 18". 

11.  a  =  5,   6  =  7,    <?  =  8. 

12.  a  =  341,    6  =  260,    e=158. 
^rXjLl3.  a  =  40,   6  =  50,   e=60. 

\r^       14.   a  =  0.654,   6  =  1?5876,   c=  0.998. 
v:       ^15.   a  =998.46,   6  =  1004.5,    c=  1268.7. 

Find  the  area  K  in  the  following  triangles : 

16.  6  =  96,   c  =  108,   ^  =  65°  10'. 

17.  a  =  480.6,   6  =  396.4,    (7=  110°  20'. 
)li8.   6  =  494,   ^  =  114°,    (7=36°. 

19.  e  =  493,    a  =  540,   A  =  76°  40'  10". 

20.  a  =  58,   6  =  65,    c=  87. 


§53]  SOLUTIONS   OF   TRIANGLES   IN   GENERAL  83 

21.  a  =375.4,   ^»  =  460.24,    c  =  584.36. 

22.  a  =  46.2,   J  =  65.8,   «=75. 

23.  ^»  =  137.6,    e=  184.5,   ^  =  110°  24'. 

24.  a  =149.24,   ^=65°  20',   ^  =  37°  28'. 

25.  To  find  the  distance  from  a  point  A  to  >S',  a  base  line 
AB  =  250  ft.,  and  the  angles  SAB  =  65°  10',  SB  A  =  48°  20'  are 
measured.  Find  AS  and  the  shortest  distance  h  from  S  to  the 
line  AB.  Ans.  AS  =  203.65  ft. ;  h=  184.82  ft. 

26.  ^  is  a  point  3  mi.  due  north  of  B ;  from  A  a  point  P 
bears  N.  68°  40'  E.,  and  from  B  it  bears  N.  35°  50'  E.  Find 
AP,  and  the  area  of  the  triangle  APB  in  square  miles. 

Ans.   3.239  mi.;  4.5268  sq.  mi. 

27.  Find  the  sides  of  a  parallelogram,  when  a  diagonal 
14.69  ft.  long  makes  angles  of  35°  48',  64°  27'  with  the  sides. 

Ans.  8.732  ft.;  13.468  ft. 

28.  A  boat  is  steaming  N.E.  at  a  rate  of  15  mi.  per  hour. 
At  10  o'clock  a  lighthouse  bears  N.  10°  W. ;  at  12  o'clock  it 
bears  W.  31°  S.  Find  the  distance  of  the  boat  from  the 
lighthouse  at  10  o'clock.  Ans.  7.774  mi. 

29.  The  line  ^^=1460  ft.  upon  shore  subtends  64°  20'  at 
a  lighthouse  which  is  985  ft  from  A.  Find  the  distance  of  the 
liglithouse  from  ^.  Ans.   1585.7  ft. 

30.  A  side  and  diagonal  of  a  parallelogram  are  690  ft.  and 
1248  ft.  respectively.  If  the  angle  between  the  diagonals 
opposite  the  given  side  is  124°  10',  what  is  the  length  of  the 
other  diagonal  ?  Ans.  214.635  ft. 

31.  To  measure  across  a  barrier  from  AtoB'a,  station  C  is 
taken,  and  the  distances  CA.,  CB,  and  the  angle  ACB  are 
found  to  be  584.6  ft.,  796.5  ft.,  and  87°  40',  respectively. 
Find  the  distance  AB.  Ans.  968.64  ft. 

32.  Find  the  length  of  a  straight  wall  which  subtends  an 
angle  of  110°  45'  at  a  point  P,  the  distances  of  P  being  487.5 
ft.  and  746.4  ft.  from  the  ends  of  the  wall.        Ans.  1025.94  ft. 

33.  To  measure  an  inaccessible  distance  XY  a  base  line 
AB=bbi)  ft.  is  laid  off  and  the  angles  ABY=  75°  48',  ABX 
=  48°  25',  5^y=58°  20',  BAX=m°  24'  are  determined. 
Find  the  length  XF.  Ans.  612.05  ft. 


84  PLANE   TRIGONOMETRY  [§53 

34.  If  the  sides  of  a  triangle  are  to  each  other  as  5  :  7  :  9, 
how  do  the  angles  compare? 

35.  The  sides  of  a  triangle  are  15,  18,  21.  Find  the  length 
of  the  perpendicular  from  the  smallest  angle  upon  the  opposite 
side.  Ans.  17.636. 

^/^  36.  Find  the  area  of  the  circle  circumscribing  the  triangle 
whose  sides  are  10,  20,  25.  Ans.  514. 

37.  From  the  ridge  of  a  mountain  range  the  depression 
angles  of  the  sides  are  55°  40',  68°  20'  respectively,  and  the 
corresponding  distances  from  the  ridge  to  the  ends  of  a  tunnel 
below  are  3475  ft.  and  2896  ft.  Find  the  length  of  the  tunnel 
through  the  mountain.  Ans.  3034.4  ft. 

38.  Show  that  the  median  drawn  to  side  a  of  the  triangle 
^-^   whose  sides  are  a,  h,  c  is  given  by  w  =  V|^(62  ^  c^)  —  i  a^. 

v^  39.  A  grass  plot  in  the  form  of  a  triangle  has  its  sides 
48.5  ft.,  65.4  ft.,  84.2  ft.,  respectively.  Find  the  radius  and 
area  of  the  largest  circular  bed  that  can  be  made  in  the  plot. 

Ans.  15.969  ft.;  801.12  sq.  ft. 

40.  The  sides  of  a  triangle  are  AB  =  24.5  ft.,  BC=  30  ft., 
CA  =  36.5  ft.  If  A,  B,  C  be  located  upon  level  ground  and 
vertical  posts  be  erected  to  heights  J.Pj  =  12  ft.,  BP^  =  18  ft., 
CPq  =  12  ft.,  what  is  the  area  of  the  triangle  P^P^P^  formed 
by  the  tops  of  the  posts?  Ana.  381.15  sq.  ft. 

41.  A  triangle  6  =  50  ft.,  ^  =  65°  24',  (7=  76°  28',  rests 
with  its  base  J  on  a  horizontal  plane,  the  plane  of  the  triangle 
being  inclined  to  the  horizontal  at  an  angle  48°  49'.  Find  the 
orthogonal  projection  of  the  triangle  upon  the  horizontal. 

Ans.  1178.3  sq.  ft. 

42.  A  parallelogram,  whose  sides  are  48  ft.,  27  ft.,  and  in- 
cluded angle  58°  48',  rests  with  its  longer  side  upon  a  hori- 
zontal plane  and  its  own  plane  inclined  to  the  horizontal  at  an 
angle  65°  50'.  Find  its  area,  and  the  area  of  its  orthogonal 
projection  upon  the  horizontal. 

Ans.  1108.5  sq.  ft.;  453.8  sq.  ft. 

43.  From  the  top  of  a  lighthouse  160  ft.  high,  the  depres- 
sion angle  of  a  ship  at  A  is  15°  48',  one  hour  later  its  depres- 
sion angle  at  a  point  B  is  10°  25',  and  the  horizontal  angle 


§53]  SOLUTIONS   OF   TRIANGLES   IN   GENERAL  85 

subtended   at   the   lighthouse   by  AB  is  100°  48'.     Find    the 
speed  of  the  sliip.  Ans.   1123.2  ft.  per  hour. 

44.  To  find  the  height  h  of  a  steeple  above  the  plane 
through  its  base  C,  a  line  AB  =  237.5  ft.  is  measured  and  the 
angles  CAB  =  40°  10'  15",  6^^^  =  110°  20'  10"  determined; 
the  angle  subtended  by  the  steeple  at  B  is  27°  18'.  Find  the 
height  of  the  steeple.  Ans.     h  =  160.61  ft. 

45.  In  Ex.  44  let  AB  =  a,  Z  CAB  =  a,  Z  CBA  =  /3,  and  the 
elevation  angle  of  the  steeple  from  B  be  7.     Show  that 

T,.v  «sina  ,  7  a  sin  a  , 

BV= — - ,  and  h  = -—  x  tan  7. 

sin(a  +  /3)  sin(a  +  yS) 

46.  At  a  horizontal  distance  a  from  a  tower,  the  angle  of 
elevation  of  the  top  of  the  tower  is  found  to  be  «,  the  angle 
of  depression  of  its  base  is  found  to  be  /8.  Show  that  the 
height  of  the  tower  is  given  by 

h  =  a(tau  a  +  tan  ^)  =  a  s^"(«  +  ^). 

cos  a  cos  yS 

47.  At  a  certain  point  in  a  horizontal  plane  the  angle  of 
elevation  of  a  peak  is  «,  a  feet  farther  away  and  in  the  same 
vertical  plane  the  elevation  angle  is  yS.  Show  that  the  dis- 
tance from  the  first  point  of  observation  to  the  foot  of  the  per- 
pendicular dropped  from  the  peak  to  the  plane  is 

,  _        a  tan  yS       _a  cos  a  sin  y3 
tan  a  —  tan  /9        sin(a  —  /8) 

and  that  the  height  of  the  peak  is  given  hy  h=  d  x  tan  a. 

48.  A  tower  148  ft.  high  stands  upon  the  top  of  a  hill ; 
from  a  point  1100  ft.  down  the  hill  the  tower  subtends  an 
angle  of  6°  48'  12".     Find  the  angle  of  inclination  of  the  hill. 

Ans.  21°  29'  57". 

49.  Two  triangles  are  determined  by  a  =  120,  b  =  160, 
^  =  35°  10'.  Find  the  difference  of  their  areas  without  solv- 
ing for  the  area  of  either  of  the  given  triangles. 

Ans.  7083. 

50.  A  ladder  40  ft.  long  is  set  with  one  end  at  a  point  15  ft. 
from  the   base  of   a   buttress,  the    other  end  reaches  a   point 


86  PLANE   TRIGONOMETRY  [§53 

35   ft.    up   its   face.       Find   the   angle  of   inclination    of   the 
face  of  the  buttress  from  the  vertical.  Ans.  8°  12'  44". 

51.  Find  the  length  of  a  window,  if  a  pole  resting  upon  the 
ground  and  making  an  angle  of  64°  28'  Avith  the  horizontal 
just  reaches  the  top  of  the  window,  and  when  its  base  is  moved 
20  ft.  farther  away,  making  an  elevation  angle  of  39°  40',  its 
top  reaches  the  window  sill.  Ans.   15.58  ft. 

52.  If  the  angle  of  elevation  of  a  balloon  from  a  point  A 

due  north  is  a,  and  at  the  same  instant  its  angle  of  elevation 

from  a  point  B  due  east  of  A  is  /3,  find  the  height  h  of  the 

\    ^^         •£   AT>  A  1  « sin  a  sin  /8 

balloon  it  AB  =  a.  Ans.     h  =  —  . 

Vsin  (a  +  y8)  sin  (a  —  ;9) 


CHAPTER   VIII 
INVERSE   FUNCTIONS.     TRIGONOMETRIC   EQUATIONS 
54.    Inverse  Notation.     From  Fig.  46  a,  Z  P  OX  =  (f). 


sin  (f>=  m^      cos  ^  =  Vl  —  m\      tan  (f)  = 


m 


Vl  —  m^ 


These  equations  may  be  expressed  inversely,  i.e.  solved  for  <f>,  thus: 
<f>  =  arc  sin  m  =  arc  cos  Vl  —  m^  =  arc  tan 


Vl-w2 


=  sin  ^  m  =  cos  ^  Vl  —  tn^  =  tan  ^ 
Y 


m 


Vl- 

Y 

YV.  , 

-1 

^ij>- 

P 

I 

/<K?^ 

0          1 
Y' 

rig.  46  b. 


From  Fig.  46  b,  with  ZPOX=  y}r,  we  have 
tan  yfr  =  t,     sin  i/r 


«  ,1  4-       I  1 

,       cos -Ur  =  —  .       COt'Ur  =  -, 


or      yfr  =  arc  tan  t  =  arc  sin 
=  tan~^  t  =  sin~^ 


VH-«2 


=  arc  cos  — z=^=  =  arc  cot  - 
VI  -t-  <2  « 


=  cos" 


VH-f2 


— _--—  =  cot  ^  - 
Vl  +  «2  « 


The  s3'mbols  sin-i  nu  cos^i  Vl  -  m\  etc.,  are  sometimes  called 
anti-sine,  anti-cosine,  etc..  but  a  better  reading  is  to  call  these 
ari'  fin  m,  arc  cosine  Vl  —  m^,  etc. 

From  the  above  drawings  it  is  to  be  noted  that  each  inverse 
function  lias  two  initial  or  primary  values.     Thus 
sin  (f)  =  m,  or  ^  =  arc  sin  m 


88  PLANE   TRIGONOMETRY  [§§54-55 

is  satisfied  by  the  angle  ^  or  tt  —  <^  =  ^'.     The  equation 

tan  ylr  =  t,  or  yjr  =  arc  tan  i, 

is  satisfied  by  the  angle  yfr  or  ir  •{-  ^|r  =  yfr' . 

In  addition  to  the  primary  solutions,  any  number  of  solutions 
to  a  trigonometric  equation  may  be  obtained  by  adding  any 
integral  multiple  of  2  tt  =  360°. 

Examples,     i.    sin  a;  =  |. 

Solution.  X  =  arc  sin  J  =  30^,  150°,  primary  solutions. 

X  =  2  mr  +-,    2  /JTT  +  -  TT,  multiple  solutions, 
6  6 

where  n  =  any  integer. 

2.  cosa  =  |^V2. 

Solution.  .  a  =  cos-i  ^  V2  =  45°,  -  45°, 

«=2n7r  +  ^,     2mr  —  -,     n  =  any  integer. 
4  4 

3.  tan  (^  =  Vs. 

Solution.  <t>  =  tan-i  VS  =  60°,  240°. 

^  z=  mr  +  - ,     n  —  any  integer. 

u 

55.  Inverse  Identities.  A  number  of  important  inverse  iden- 
tities will  be  introduced. 

I.    sin-^£c  +  sin-^y  =  sin-^  (xVl  —  y^+  yVl-  x-). 

This  identity  may  be  established  as  follows :  (1)  construct 
an  angle  ^  whose  sine  is  x,  also  an  angle  i/r  whose  sine  is  i/. 
(2)  The  left  member  of  the  identity  is  (fy  +  yjr.     Take 

sin  (0  +  \/r)  =  sin  ^  cos  sjr  +  cos  0  sin  ^jr 


=  X  Vl  —  ip'  +  y  Vl  —  7?. 
Hence, 

^  +  -\^  =  sin"ia;  +  sin~iy  =  sin~^  (a;Vl  —  y"^  +  y^l  —  oiF). 
In  a  similar  manner 


sin  ^  X—  sin~^  ?/  =  sin"^  {x  Vl  —  y^  —  y  Vl  —  a;^) . 
II.    cos-^cc  ±  co^-^y  =  cos"^  {xy  T  VCl  —  £c"'^)(l  —  y-)). 

This  identity  may  be  established  by  constructing  two  angles, 
^  =  \3os~^  a;,  i/r  =  cos~^  y,  taking  the  cosine  of  the  sum  and  differ- 


§55]  INVERSE   FUNCTIONS  89 

ence,  substituting  as  above,  and  then  passing  to  inverse  no- 
tation. 

Prove  the  following : 

III.  tan-'a;±tan-iy  =  tan-'f5''=^  V 

IV.  Hin-i  x  =  2  8in-»  (2  a? vl  -  a;'^)  =  -  »ln-^  i-  x). 
V.    co8-Ja5  =  |co»-»(2ic2- 1). 

EXERCISES 

Construct  the  acute  angles  indicated  in  the  inverse  notation, 
and  find  values  of  the  following : 

1.  cos(8in~^^). 

Suggestion.     Construct  <(>  —  nin-^d),  then  co8<^  =  jV3. 

2.  tan(8in-i|).  9.    co8(90°  —  cos-^a;. 

3.  tan  Care  cot -M-)'  /-  tt 

_  10.    8in(tan-iV8)-co8^. 

4.  cot(  arcsin— -  )• 

5.  sin  (arc  tan  1). 

6.  cos(arccot  0). 

7.  tan  (arc  cot  1). 

8.  sin (90° -sin -11), 

Verify  the  following : 

15.  sin"i  I  +  cos'i  I  = 

16.  arc  tan  1  +  arc  cos 

17.  arc  sin  1  —  arc  tan  1  =  45". 

18.  arc  cos  1  +  arc  tan  qo  —  arc  cot  1  =  45°. 

19.  arc  vers  1  —  arc  sec  V2  =  45°. 

20.  sin  (90°  -  tan-i  V8)+  tan(90°  -  sec'i  V2)  =  |. 

21.  arc  sin  '-  +  arc  cos  -  =  —  •        23.    arc  tan  -  +  arc  tan  -  =  — . 

5  5      2  b  7      4 


11. 

sin  (2  cot- V3). 

12. 

co8(3  8in~i|). 

13. 

tan  (90°- sec- ^2). 

14. 

co8(90°-8in-4f). 

90°. 

] 

V2 

=  90°. 

22.    arc  sin  a;  4-  arc  cos  a;  =  — . 


24.    sinf  2arc8in-j  =  -V3, 


90  PLANE   TRIGONOMETRY  [§§  55-56 

25.  arc  tan  -  +  arc  tan  -  =  arc  tan  1  =  —• 

26.  arc  tan =  2  arc  tan  x  =  arc  sin 


27.    sin~i  x= cos~^  ^  ~  o  ^^^~^  (1  ~  -  ^)* 


28.  cos~^a:  = sin  ^a;=  2  tan'^xl 

2  ^l-\-  X 

29.  sm~^(^^x  —  4:a^)  =  dsm~^x. 

30.  cos~^(4a:3  — 3a;)=  3cos~^2;. 

31.  tan-iP^~^=  Stan-Ire. 

\l-SxV 


S.    tan-if-^Vtan-i^:tl  =  tan-i('-i-Y 


32. 

1 


33.  2sin-ia;=  sec-1 

1-22^ 

34.  tan~i —  =^ cos-ia;2_;_gin-i2:;2^ 

Vl+a:2  +  Vl-a.-2      4-2  2 

TRIGONOMETRIC   EQUATIONS 

56.  Definitions.  An  equation  containing  07ie  or  more  trigo- 
nometric functions  of  an  unknown  angle  is  called  a  trigonometric 
equation.     Thus, 

(a)    sin  a:  =  |, 

(J)   tan  X  -{-  sin  x  =  5,  f    sin  a;  +  cos  t/ =  ^, 

^  X      .         ,         X     a  1  Ssinaj— cos  y  =  1, 

(c)    sin  X  +  cos  ^  =  ^1  ^        ' 

are  trigonometric  equations.     Equations  (<?)  above  are  simultane- 
ous trigonometric  equations. 

The  operations  of  ordinary  algebra  —  clearing  of  fractions, 
transposing.,  multiplying  hy  constants  —  are  applicable  to  trigo- 
nometric equations. 

In  addition  to  these  operations,  the  transformations  of  trigo- 
nometric identities  may  also  be  brought  into  use.  For  exam- 
ple, the  equation 

sin  2  a;  =  cos  x 


§§57-58]  TRIGONOMETRIC   EQUATIONS  91 

may  be  changed  trigonometrically  into 

2  sin  X  cos  x  =  cos  x. 
then  transpose  and  factor, 

cos  a;(2  sin  a:  —  1)  =  0. 
Equating  to  zero  each  factor, 

cos2;=0,   a;  =90°,  or  270\ 
2sincc  =  l,    a;  =30°,  or  150°. 

57.  Solutions.  Trigonometric  equations  differ  from  algebraic 
equations  in  one  important  particular,  viz.  they  have  a  multi- 
tude of  solutions^  whereas  algebraic  equations  have  a  finite  num- 
ber of  solutions.     As  illustrations  notice  the  following  examples. 

(1)  sin  a:  =  1  has  x  =  30°,  150°,  or  a:  =  2  wtt  +  -,  or 

2  6 

(2w  +  l)7r-J,    (ri  =  0,  ±1,  ±2,  ...). 
b 

(2)  tana:  =  V3  has  for  solutions  a;  =  60°,  240°,  or   W7r  +  — , 

o 

when  n  =  any  integer. 

We  shall  enumerate  and  illustrate  some  of  the  more  impor- 
tant types  of  trigonometric  equations. 

58.  Simple  Equations.  Under  this  heading  may  be  included 
any  equation  which  reduces  readily  to  one  of  the  forms : 

^  ^  ~"     '  i  where  ^is  not  grreater  numerically  than  1. 

(2)  cos  a;  =  ^,  J  ^  ^ 

(3)  tan  x  =  K,  K=  any  number. 

Examples,     i.    Solve  4  sin  a;  =  esc  x  f^r  the  angle  x. 

Solution.     Multiply  by  sin  z,  and  divide  by  4,  giving 

siu'-^  X  =  \, 
or  sin  a;  =  ±  ^. 

Hence,  x  =  sin-i( ±  i)  =  30°,  -  .30° ;  150°,  -  150° ; 

or,  in  general  notation, 

X  =  mr±  --,  n  =  0,  1,  2,  3,  •  ••. 
o 


92  PLANE   TRIGONOMETRY  [§58 

2.  Find  X  from  tan^a:  +  4  =  2  sec^a;. 

Solution.     Replace  sec'^  x  by  1  +  tan^  x,  transpose,  collect,  and  change  signs, 
tan^  X  =  2,  or  tan  x  =  ±  V2. 

Then, 

X  =  tan-i(±  V2)  =  tan-  (±  1.4142)  =  54°  44',  -  (54°  44'), 

or  X  =  nTT  ±(54"  44'). 

Note.     Take  the  angle  tan-^(V^)  from  the  table  of  natural  functions. 

3.  Find  X,  when  6  cot  x  +  5  =  tan  x. 

Solution.     Multiply  by  tan  x,  transpose,  and  change  signs, 

tan^  X  —  5  tan  x  —  6  =  0, 

a  quadratic  equation  with  tan  x  as  the  variable,  which  factors  into 

(tan  X  +  l)(tan  x  —  6)  =  0, 
giving  j  tan  x  =  —  1, 

and  I  tan  x  =  6. 

Hence,  Xj  =  135°,  -  45°, 

or  X2  =  tan-i  6  =  80°  32',  260°  32'. 

The  general  values  of  x  are 

xi  =  nTr  -  ^,  xo  =  mr  +  80°  32'. 

EXERCISES 

Solve  the  following  trigonometric  equations,  giving  only  the 
solutions  which  are  between  —  180°  and  +  180°,  inclusive. 

1.  3  sin  a;  =  2.        Ans.    sin-^  f  =  41°  48'  35",  or  138°  11'  25". 

2.  sin  2x  —  cos  a:  =  0.      [sin  2  a;  =  2  sin  x  cos  x.  ] 

Ana.    ±  90°,  30°,  150°. 

3.  COS  2  a;  +  sin  a:  =  1.  Ans.    0,  — ,  ,  tt. 

4.  COS  2  a:  =  sin  x.    •  Ans.    30°,  -  90°,  150°. 

5.  tan  2  a:  -f-  2  sin  a;  =  0.  Ans.    0,  ±  60°,  180°. 

6.  COS  3  a;  —  sin  2  a;  =  0. 

Suggestion.  Change  cos  3  x  to  4  cos^x  —  3  cos  x,  sin  2  x  =  2  sin  x  cos  x, 
giving  4  cos^  x  —  3  cos  x  =  2  sin  x  cos  x ;  factor  and  solve.  Or,  cos  3  x  = 
sin(3  X  +  90°)  ;  then 

cos  3  X  —  sin  2  x  =  sin  (3  x  +  90°)  —  sin  2  x 


=  2cosf|x  +  45°)  •  sinf|+  45°)  =  0. 


§§58-59]  TRIGONOMETRIC   EQUATIONS  93 


Hence, 

008(2X4-  45°)  =  0,  and  sinf-4 

giving 

^  a:  +  45°  =  90",  270^  450'"', 

and 

|  +  45°  =  0°,  180°. 

Then, 

X  =  18°,  90°,  162", 

and 

X  =  -  90°. 

7.  COS  3  a:  +  sin  2  a; -COS  a;  =0.     Ans.   0°,  30°,  90°,  150°,  180°. 

8.  cos(a:  +  60°)  -  sin(a;  +  30°)  =  1 V3.    Ans.    -  30°,  -  150°. 

9.  sin(a;  +  60°)-sin(a;-60°)  =  -lV3.  Ans.    ±120°. 
10.  sin  4  a;  =  2  sin  2  a;.                                    Ans.    0°,  90°,  180°. 

59.    Equations  of  the  Form 

r  cos  (|>  =  a, 
r  sin  <J)  =  6. 

Here  is  a  set  of  two  simultaneous  equations  in  two  unknowns, 
r,  <^. 

(1)  To  find  ^,  divide  the  second  equation  by  the  first, 

tan^  =  -,    ^  =  tan~i(- 
a  \a. 


Or,  ^  =  HTT  +  tan  ^(-j- 


(2)  To  find  r,  square  and  add,  recalling  the  identity  sin^  <^  + 

cos^^  =  1, 

7^  =  a^  +  b\ 


r=±Va2  +  p. 

The  proper  sign  of  r  must  be  chosen  so  that  r  cos  ^  =  a, 
r  sin  <f)  =  b. 

Examples.     1.    Find  r,  <f>  from 

fr  cos  ^  =  3, 
r  sin  (f>  =  4. 

We  have  tan<^  =  |,    <^  =  tan-i(f )  =  53°  8'. 

Squaring  and  adding,     r^  =  4^  4-  3^  =  25, 

r  =  ±  5. 


94 


PLANE   TRIGONOMETRY 


[§§  59-60 


Find  r,  (f>  in  the  following 


2. 


3. 


rcos<^  =  12, 
r  sin  (f>  =  5. 

r  cos  <f>=  Q, 
rsin  ^=  12. 


r  sin  0  =  5, 
rcos<^  =  5V8. 

/•sin<^  =  4.876, 
r  cos  i  =  2.396. 


60:    Equations  in  the  Form 

r  sin  6  cos  <|)  =  a, 

r  sin  6  sin  (j>  =  6, 

r  cos  9  =  c, 

where  r,  0,  ^  are  variables. 

Divide  the  second  equation  by  the  first, 

tan0  =  -,    <f>  =  tan~^-' 
a  a 

Squaring  all  three  equations  and  adding,  we  have 

r2  (sin2  d  [cos2  </>  +  sin^  (f>-\  +  cos^  e)=a^  +  b'^+  c^, 
or  r^  =  a^  +  P  +  c^, 

r=±  Va2  +  J2  4.  c2. 
From  the  third  equation, 

cos  t/  =  -  = 


r       -t  V«2  +  62  4.  ^2 

^=:COS-lf ^^ Y 

V V«2  +  j2  .  ^y 


EXERCISES 

Find  r,  ^,  </>  in  the  following : 

r  sin  ^  cos  0  =  2, 
1.    ^  r  sin  0  sin  0=2,  4. 

rcos^  =  l. 

r  sin  ^  cos  0  =  6, 
/•  sin  ^  sin  0  =  3,  5. 

r  cos  0=2. 

r  sin  ^  cos  0=2, 
r  sin  ^  sin  0  =  6,  6. 

r  cos  ^  =  9. 


r  sin  ^  cos  0  =  12, 
rsin  ^sin0  =  12, 
r  cos  ^  =  1. 

r  sin  ^  cos  0  =  1, 

r  sin  ^  sin  0=4, 

r  cos  ^  =  8. 

r  sin  0  cos  0=5, 

r  sin  ^  sin  0  =  2, 

rcos^=  0. 


§§Gl-62]  TRIGONOMETRIC   EQUATIONS  95 

61.    To  solve  a  sin  jr +  6cos  jr  =  c.     Divide  this  equation  by 

.^^  a    sin  X       b  •  cos  z  c 


Now  let        — r==.  =  sin  <f),  — rr=  =  COS  6, 

Va2  +  62  Vrt2  +  62 

and  hence,  -  =  tan  </>,        0  =  tan"^  -  • 

6  6 

Then  equation  (1)  becomes 
(2)   sin  ^  sin  x  -f  cos  ^  cos  x  = 


or  cos  (a;  -  6)  =        ^ —  *  a;— rf>  =  ±cos  ^( —  ). 

Va^  +  62  ^  Wa2  +  62/ 


c 


X  =  (b  ±  cos  ^        

\Va2  -f  62> 

Examples,     l.    Solve  5  sin  a;  +  12  cos  a:  =  6.5. 
Divide  both  members  by  13, 

(1)  j^  sin  ^  +  yf  cos  re  =  0.5. 

Write  ^^  =  sin  <^,    1|  =  cos  <f>,    tan  ^  =  ^^3. 

Hence,  </)  =  tan-i(-^2)  =  22°37'. 

Now  equation  (1)  becomes 

(2)  sin  <f>  sin  x  +  cos  <^  cos  a;  =  0.5, 
or  cos  (a:  —  </>)  =  0. 5, 

x  =  <f>±  cos-i(0.5)  =  22°  37'  ±  60°  =  82°  37',  -  37°  23'. 

2.  Solve   3  cos   x  +  5  sin  x=  4. 

3.  Solve  12  sin  a;  +  5  cos  a;  =  3.9. 

4.  Solve  8  cos  a;  +  15  sin  a;  =  5.1. 

5.  Solve  3  cos  a;  —  2  sin  a;  =  I  Vl3. 

6.  Solve  5  sin  x  —  6  cos  x  =  ^  V61. 

62.    To  solve  sin  (;r  +  4>)  =  a  sin  r. 

We  have  sin  (a:  +  cf. )  ^  a 

sin  X  1 


96  PLANE   TRIGONOMETRY  [§§62-63 

Take  by  composition  and  division, 

sin  {x+  (f>}  +  sin  x  _a  +  1 
sin  (a;  +  <^)  —  sin  x      a  —  1 ' 

d)\    .     (b      a—  1 


or  tanlic  +  f  )  =  ^^tan|' 

2/      a—  1         2 


The  right  member  is  now  known,  and  the  solution  may  be 
written  out. 

Examples,     i.    Solve  sin  (x  +  37°  14')  =  2  sin  x. 

Substituting  in  the  above  result,  and  retaining  the  smallest 
angle, 

tarifx+^^^^)  =  Stan  ^^^^=  3(0.3369), 

2:=tan-i(1.0107)-^^^^=45°18'20"-(18°37')  =  26°41'20". 

2.  Solve  sin  (x  +  65°  21')  =  3  sin  x. 

3.  Solve  sin  (x  -  28°  40' )  =  |  sin  x. 

4.  Solve  sin  (a;  +  56°  24')  =  5  sin  (x  -  10°  20'). 

5.  Solve  sin  (x  +  94°  10')  =  4  sin  x. 

6.  Solve  sin  {x  —  124°)  =  |  sin  x. 

63.    To  solve  tan  (jr  +  <}))  =  fl  tan  x. 

Divide  by  tan  a;,  and  take  by  composition  and  division, 

tan  (a;  +  ^)  _ 
tan  X  ' 

tan  (x+  (f)')  +  tan  x      a+1 
tan  (a:  +  0)  —  tan  x      a  —  1 

o-       Tc  •  sm(2x+  <b)      a  +  1 

Simplifying,      \     j         =  -^ ' 

sin  (p  a  —  1 


§§63-64]  TRIGONOMETRIC   EQUATIONS  97 

Examples,     i.    Solve  tan  (x  +  20°)  =  5  tan  x. 
Comparing  with  the  above  result, 

4>  =  20°,  a  =  5, 
sin  (2  a;  +  0)  =  I  sin  20° 

=  1(0.342)  =0.513, 
2x  +  (t)  =  sin-i  (0.513)  =  30°  52', 
2  a;  =  30°  52' -20°  =  10°  52', 
a:  =  5°  26'. 

2.  Solve  tan  (x  +  30°)=  6  tan  x. 

3.  Solve  tan  (x  +  47°  20')  =  7  tan  x. 

4.  Solve  tan  (x  +  25°  10')  =  10  tan  x. 

5.  Solve  tan  (x  +  60°)  =13  tan  x. 

64.    To  solve  jr  =  a  +  p  sin  jr. 

In  this  equation  a,  ^  are  usually  given  as  angles  (degrees  or 
radians)  ;  /9  expressed  in  radian  measure  is  smaller  than  unity. 
Two  plans  of  solution  will  be  sketched. 

(1)  Trial  solution.  Let  a,  /8  be  expressed  in  degrees.  Then 
an  upper  limit  to  x  will  be  shown  by  a  +  /3,  since  the  multiplier 
yS  is  smaller  than  unity.  Take  a  trial  solution,  substitute  in 
the  equation,  note  the  error,  make  another  approximation,  and 
so  continue  until  the  required  degree  of  accuracy  is  attained. 

(2)  G-raphical  solution.  Let  a,  /3  be  expressed  in  radian 
measure  -.  a°  =  a  radians,  /3°  =  6  radians.  Then  we  have  to 
determine  x  so  that 

X—  a=h  sin x. 
Let  y^  =  x  —  a,     yi=^  sin  x. 

Construct  upon  rectangular  axes  a  curve  representing  each  of 
these  equations.  The  first  is  a  straight  line  through  the  point  a 
on  the  X-axis  ;  the  second  is  a  modified  sine  curve.  The  x  of  the 
point  of  section  of  these  two  graphs  is  the  required  solution. 

As  an   illustration   of   the    graphical    solution    let  us  solve 
X  =  57°  17'  44"  +  45°  x  sin  x. 
Here,        57°  17'  44"  =  1  radian,     45°  =  0. 785  radians. 
Then  we  are  to  solve  x  =  l  -h  0.785  x  sin  x. 

Let  3/j  =  a;  —  1,     ^/g  =  ^•'^^^^  sina;. 


98 


PLANE   TRIGONOMETRY 


[§64 


Now  construct  the  straight  line  y^  =  x—\^  and  the  modified  sine 
curve  ^2  =  0.785  sin  x.  The  abscissa  of  the  point  of  intersection 
of  the  straight  line  and  curve  is  approximately  x  =  1.769  radian 
=  lOr  27'. 

Y 


Fig.  47. 

Examples,     i.    Solve  a;  =  24°  +  30**  x  sin  x. 

2.  Solve  a:  =  64°+  28°  x  sin  x. 

3.  Solve  X  =  30°  +  50°  x  sin  x. 

4.  Solve  X  =  10°  20'  +  40°  x  sin  x. 

5.  Solve  a:  =  45°  +  (37°  30')  x  cos  x. 

6.  Show  sin  a; ^— a;,     O^a;^^- 

TT  2 


CHAPTER   IX 

COMPLEX  NUMBERS.  DEMOIVRE'S  THEOREM.  TRIGONO- 
METRIC SERIES.  EXPONENTIAL  AND  HYPERBOLIC 
FUNCTIONS 

65.  Roots  of  Quadratic  Equations.  In  ordinary  algebra  we 
have  such  equations  as 

x^—  16  x+  25  =  0, 
whose  roots, 

are  called  coihplex  numbers.     These  numbers  contain  a  real  unit, 
1,  and  a  so-called  imaginary  unit,  V—  1. 

CI)  Properties   of  V— 1.      The   imaginary   unit   is   usually 

represented  by  i.     We  may  easily  show  that  when  V—  1  =  *, 

2*2  =  —  1,  1^  =  —  i,  t*  =  1,  i^  =  i  •••, 
and  generally, 

z4*  =  1,    |4^-+1  =  i,    i4*+2  =  _  1,    i4A-+3  =  _i^ 

(2)  Graphical  representation  of  x  +  yi.  To  any  complex 
number  as  3  +  4  i,  8—4  i,  x  +  yi  corresponds  a  point  in  a  plane. 
If  we  multiply  a  real  number  a  by  i,  and  this  product  again  by 
i,  the  result  is  —  a.  Thus,  multiplying  twice  by  i  changes  a 
number  to  its  negative.  Multiplying  a  number  by  i  may  be  in- 
terpreted as  turning  its  direction  through  90°.  To  locate  3  +4  i 
upon  a  plane,  lay  off  3  units,  OQ,  along  the  real  axis  (horizon- 
tal in  Fig.  48),  then  at  Q  erect  a  perpendicular  4  units  long; 
the  point  Pj  represents  the  complex  number  3  -f  4  i.  Lay  off 
—  4  perpendicular  to  OQ  at  Q,  and  we  locate  3  —  4  t,  at  P^. 

Any  complex  number  x  -f-  yi  may  be  represented  upon  a  plane 
as  shown  in  Fig.  49.  The  point  P  may  be  anywhere  in  the 
plane. 

(3)  Modulus,  arc  (re  -f-  yi').  The  line  OP  =r  is  a  vector  equal 
in  length  to  the  modulus  of   x  +  yi,  or  the    absolute   value   of 

99 


100 


PLANE  TRIGONOMETRY 


[§§  65-66 


X  +  yi.     The  angle  XOP  =  <^  is  called  the  arc  of  a;  +  yi,  or 
amplitude^  or  argument  of  re  +  yi.     As  abbreviations 

r  =  mod  (x  +  yi),  ^  =  amp  (a?  +  yi). 


-X- 


Fig.  48. 

The  following  notation  should  be  recognized : 


Fig.  49. 


Modulas  of  x  +  yi  =  r  =  Vx'^  +  y^  =  \x  +  yi\. 

Arc  of  a?  +  2/i  =  <|)  =  tan-^  ^  =  amplitude  (x  +  yi). 

EXERCISES 

1.  Locate  the  following  complex  numbers  : 

(1)  3  +  2t;   (2)  2  +  6^;   (3)    -S  +  Si;   (4)    -4  +  z; 

(5)   -3-42;   (6)   -6i;   (7)  4-5i;   (8)  (^l+^z^x5. 

2.  Find  the  modulus  and  arc  of   each  of   the  numbers   in 
Ex.  1. 

3.  Solve  the  following  equations  and   locate  the  roots  as 
complex  numbers : 

(1)  22_43  +  13  =  0;  (2)  z2  +  62  +  13  =  0;  (3)  22  +  ^  +  1  =  0; 
(4)  z^-l=0;   (5)  z3+l  =  0. 

4.  Locate  the  following  products  :   (1)  t  x  (2  +  4  i) ; 
(2)  ix(i-S  +  2i);   (3)  ^x(5-3^);  (4)  ixixi(2-i). 

66.    Complex   Numbers   expressed   Trigonometrically.     From 
Fig.  49,  we  have 

X  =  rcos <|>,  y=r sin <|>,  r  =  Va?^  +  y-,  <j>  =  tau"^ ^- 


§§  66-67]  DE  MOIVRE'S   THEOREM  101 

Hence, 


a;  +  yi  =  r(cos  (|)  +  i  sin  <}>)  =  vxM-j^(cos  <j>  +  i  sin  <j)). 

Theorem.     A  complex  number  equals  its  modulus  multiplied 
by  the  expression  cos  <^  +  ^  sin  (f>  ivliere  ^  is  its  amplitude. 

Examples,     i,   2^  —  2  +  1  =  0  has  roots, 


1  ,  V8  .       _  1      V8  . 
2"^  "2"''  ^2~2~^*" 


The  modulus  of  z-^  =  1,  mod  z^=\',  the  amplitude  of 

z^  =  tan-'^^  ^  ^^  =  tan-i VB  =  60°, 

am  Z2  =  tan-i(  -  V3)  =  -  60°. 
Hence, 

2,  =  -  +  —  ^  =  cos  60°  +  i  sin  60°, 
^22 

2,  = i  =  cos  60°  —  i  sin  60°. 

2     2        2 

2.    Express  the  roots  of  the  following  equations  in  trigono- 
metric form  : 

(1)  22  +  1  =  0  ;     (2)  22  -  2  2  +  2  =  0  ;     (3)   22  -  V3  2  +  1  =  0  ; 
(4)  22  +  2  2  -  8  =  0  ;   (5)  22  +  V2  2  +  1  =  0  ;   (6)  22  +  2  +  1  =  0. 

67.   DeMoivre's  Theorem.     Let  us  take  a  complex  number  in 
trigonometric  form, 

z  =  X  +  yi  =  r(cos  <^  +  i  sin  0). 

Squaring  2,  and  recalling  that  1*2  =  —  1, 

^2  _  7.2(^(3Qg  (^  -\-  i  sin  ^)2  =  r2(cos2  ^  —  sin2  <^  +  2 1  sin  <^  cos  ^), 

(1)  22  =  r2(cos  2  </)  +  z  sin  2  <^),  §  35,  (7),  (8). 
Now,  multiply  (1)  by  2  =  r(cos  <^  +  i  sin  ^). 

2^  =  r^[cos  2  (/)  cos  <^  —  sin  2  «^  sin  <^ 

+  z(sin  2  <f>  cos  <^  +  cos  2  ^  sin  <^)] . 

(2)  23=/-3(cos3</)  +  isin3</>),  §34. 


102  PLANE   TRIGONOMETRY  [§§  07-68 

The   law  of   exponents   shown    in  (1),  (2)  would   indicate 
that  for  n  =  any  positive  integer, 

(3)  2"  =  r"(cos  n^  +  i  sin  w^). 

Assuming  law  (3)  to  hold  for  any  integral  w,  let  us  see  if  it 
holds  when  n  is  replaced  by  n  +  1. 
Multiply  (3)  by 

2  =  r(cos  4>  +  i  sin  ^), 
2"+i  =  r"+i  [cos  n^  cos  ^  —  sin  n^  sin  ^  +  i(sin  n^  cos  ^ 
+  C0S  w^sin^)]. 
(4)  2"+i  =  r"+i[cos  (w  +  1)0  +  i  sin  (w  +  1)<^] . 

Hence,  the  law  assumed  in  (3)  for  the  integer  n  holds  for  n  +  1 . 
We  see  this  law  holds  for  n  =  2  and  w  =  3,  hence  it  holds  for 
w  =  4,  5,  •••,  n=  any  positive  integer.     Hence,  we  have 

DeMoivre's  Theorem:   (cos <}>  +  i sin <{))»*  =  cos w(})  +  * sin /kJ). 

DeMoivre' s  Theorem  holds  when  n  is  an  integer,  a  fraction,  or  a 
negative  number. 

(cos<t>  +  *8in<j>)»i  =  COS  — +  i8in  — , 
n  n 

p  p  p 

(co8<|)  +  i  sin  <{>)«  =  cos— <j)  +  isin— <j), 

(cos  <J>  +  i  sin  <j))  ~*^=  cos(—  ?w<|))  +  i  sin(—  wit})) 
=  cos  wi(j)  —  i  sin  m^, 

68.    Raising   to  Powers  and  Extracting   Roots.     DeMoivre's 
Theorem  enables  us  to  raise 

z  =  x-\-  iy  =  r(cos  ^•\-i  sin  0) 

to  any  power,  or  to  extract  any  root  of  z. 

Thus,  z^  =  r\Gos  2<f>  +  i  sin  2  <^). 

Hence,  to  square  a  complex  number,  square  its  modulus  and 
double  its  amplitude.  To  cube  a  complex  number,  cube  its 
modulus  and  multiply  its  amplitude  by  three. 


§68]  DEMOIVRE'S   THEOREM  103 

Examples,     l.    Raise  z  =  3  +  4 1  to  the  2d  power  ;    to  the 
3d  power. 

2   =  3  +  4  I  =  5(cos  <^  +  i  sin  <f>),  (f>  =  tan-^  *  =  o'd^  8',  nearly. 
22  =  (3  +  4  1)2  =  25(cos  2  ^  +  J  sin  2  <^). 
z^=  (3  +  4  0^  =  125 (cos  3  <^  +  i  sin  3  <^). 

2.  Raise  z  =  -  +  —^i  to  2d,  3d,  rth  powers,  and  locate  these 
respective  numbers  on  a  diagram.     See  Fig.  49. 

3.  Find  ^^  2^,  2^,  when  z  =  — \-  -i. 

4.  Find  z^,  z^^,  when  z  =  —  1  —  i. 

The  extraction  of  roots  may  be  performed  by  use  of  DeMoivre's 
Theorem : 

zn  =  r«[  COS  —  +  t  sin  ^ 
\      n  n 

This  formula  seems  to  give  but  one  of  the  n  wth  roots  of  2, 
but  we  may  obtain  n  different  roots  by  writing 

2  =  /•[cos(^  +  2  kir^  +  i  sin(^  +  2  Attt)]  ; 


then  will 

1  ir  J^    1     O  7 J.    1     O  7 " 

,  A:  =  1,  2,  3,  •••,  n—\. 


1^        1 

2»  =  T" 


cos-^— ' h  1 2— ' 


n  n 

Examples      i.  Extract  the  square  root  of  i. 
Here,  z  =  i  =  \{ cos-  +  i  sin ^  1 

=  cosf^  +  2  A-tt]  +  I  sin  ( I  +  2  i-7r\ 

r        1             /5+2^^\              /|  +  2A-,r\ 
Then,  z-  =  i^  =  cosf  ^^^ j  +  i  sin(  ^— ^ )• 

Let  zi,  Zj  be  the  two  roots,  then 

2,  =  COS-+ isin^  =  — +  — t,  *  =  0, 

^  4  4      V2      V2 

2„  =  cos  ( ^  +  TT  )  +  t  sin  (  ^  +  tt]  = -i,  k  =  \. 

^  \4         y  V4         /  v''2      V2 


104 


PLANE  TRIGONOMETRY 


[§§  68-69 


2.    Extract  the  cube  root  of  —  8,  and  locate  the  roots  on  a 
diagram. 

In  this  case    z  =  —  8  =  8(cos  tt  +  ^■  sin  tt) 

=  8[cos(7r  +  2kTr)  +  i  sin(7r  +  2  ^tt)]. 
Extract  the  cube  root, 


z*=  (-8)^  =  8*rcos 


ir  +  ^kir 


+  «sin'^  +  2*^- 


3  3 

Giving  k  values  0,  1,  2,  we  find  the  three  roots,  Zj,  z^,  z^, 

2i  =  2(003-  +  J  sin -)  =  1  +  V3  I,       k^O, 
\        o  0/ 

Zj  =  2  (cos  IT  +  i  sin  tt)  =  —  2,  k  =1, 

23  =  2f  cos^  +  tsin5J:'j=  1  -  V3  i,  ^  =  2. 

3.    Find  the  five  5th  roots  of  32 ;  of  i. 


-l  +  V3^• 
2 


4.    Find  the  six  6th  roots  of  —  1  ;  the  cube  roots  of 
69.    Value  of  sin  jr,  cos  x  in  Terms  of  x. 
(1)    Value  of ^  wAgw  ^  =  0. 

When  an  angle  <^  is  small,  sin  <f)  approaches  arc  cf). 

From  the  tables  of  natural  functions  and  radian  measure  we 
have  : 

sin    0°  =  0.00000  0°  =  0.00000  radian 

sin  10'  =  0.00291  10'  =  0.00291  radian 

sin  40'  =  0.01164  40'  =  0.01164  radian 

sin    1°  =  0.01745  1°  =  0.01745  radian 

sin    2°  =  0.03490  2°  =  0.03491  radian 

which  show  that  sin  <f>  =  <f)  for  values  of  <f>  from  0°  to  near  2°, 
true  to  five  decimals. 

To  show  this  property  generally, 
we  have  from  geometry 

or,  if  OA  =  1,  Fig.  50, 

sin  (f)<,<f)<.  tan  <f>. 
Divide  this  inequality  by  sin  ^, 


1< 


± 
sin  <i>  "  cos  <f) 


< 


Fig.  50. 


DEMOIVRE'S   THEOREM 


105 


Now  let  <}>  approach  zero,  0  =  0,  and  cos  ^  =  1 ;    hence, 


1< 


4> 


or. 


limit 


_sin  (f) 

4> 


?^ 


=  1. 


<l, 


(2)    Value  of  sin  ncf),  cos  n<f)  in  terms  of  sin  <f>,  cos  0. 
In  algebra  it  is  shown  that  if 

X  +  ^i  =  a  +  bi, 

then  x=  a,     y  =  b. 

Theokem.     If  tivo  complex  numbers  are  equals  the  real  parts 
are  equal  and  the  imaginary  parts  are  equal. 

By  DeMoivre's  Theorem, 

(-4.)  (cos  (f)  +  i  sin  0)"  =  cos  n(f>  +  i  sin  w0. 

But  by  the  Binomial  Theorem, 

(^B)  (cos  (f)  +  i  sin  <f>')"  =  cos"  (f>  +  n  cos""!  ^  (i  sin  <^) 
+  '^^f""^^  cos«-2 (p (i sin <^)2  +  n{n-lYn-2-)  ^^^„_3 ^ ^. ^^^ ^y 

,    ?<  (7Z  —  1)(W  —  2)(w—  3)  n-iJ./"     •      JL\4    1 

1  •  2  •  o  ■  4 

the  series  terminating  if  w  is  a  positive  integer,  and  becoming 
infinite  if  w  be  a  fraction  or  negative.  Now  i^  =  —  1,  z^  =  —  i, 
i*=l,  ••• ;  hence,  the  first,  third,  fifth,  •••,  terms  of  the  right 
member  of  (B)  are  free  of  i,  and  the  even  terms  contain  i.  The 
right  members  of  (A}  and  (-B)  are  equal.  Equating  the  real 
and  imaginary  parts,  respectively,  we  have 

( (7)  cos  n<f>  =  cos"  <f> \  ~a      cos*'"^  (f)  sin^  </> 


H i;^ -^ ^ ^cos"  *  <f)  sin*  ©  —  •••, 

1  .2-3-4  ^         ^ 


106  PLANE   TRIGONOMETRY  [§69 

(i))  sin  n<f>  =  n  sin""i  <^  cos  </> T  o    o~      cos^'^  (f>  sin-'^  <^ 

1  •  2  •  o 

.  (»  -  1)0.  -  2)(«  -  3)(»  -  4)  ,        ,, 

l-2-a-4-5  ^         ^ 

If  in  (0^),  (2^)  we  give  w  the  values  2,  3,  4,  •••,  we  may 
obtain  the  ordinary  expressions  for  sin  2  <^,  cos  2  ^  ;  sin  3  0, 
cos  3  <^ ;  etc.,  in  terms  of  sin  0,  cos  </>.     See  §§35,  36. 

Examples,  i.  Show  from  ((7)  cos  2  </>  =  cos^  (/>  —  sin^  <^, 
w  =  2. 

2.  Show  from  (2))  sin  2^=2  sin  ^  cos  <^. 

3.  Show  cos  3  <^  =  4  cos^  ^  —  3  cos  (j). 

4.  Show  sin  3  </>  =  3  sin  ^  —  4  sin^  (f). 

(3)   Trigonometric  series.     In  formulas  (C),  (D)  above,  let 
us  substitute  w<^  =  a;,  or  w  =  — , 
and  we  have 

(C'}  cos  x=  cos*"  (fy-'^^f        ^  cos"-2  0  sin2  </>  +  ... 

«  ,       x(x  —  d>^       „_„  ,  /sin  (/>\2 
cos"  d> ^ — -^  cos"  2  (f,  / r     _|_  . . 


^  1-2         ^      ^l^    </>    y 


(2)')  ^^^_lY^_2 


sin  a;  =  -r  cos"~i  <f>  sin  ^  —  — ^r— ^ cos""^  d>  sin^.d>  + 

<p  1  •  !Z  •  o 

=  a;  cos-i  </.  (^  -  ^^^-^^X^-^'^)  cos-3  ^  (^^'^  +  .... 

Now  let  n=cc  in  such  manner  that  n^^x,  then  ^^ ^  =  1, 

see  (1)  above,  and  (0"),  (2)')  become  the  infinite  trigonometric 

series 

I.    e.sx  =  l-|+J-|+..., 

II.    staa,=a,-|  +  |-|  +  ..., 


§  69]  DE  MOIVRE'S   THEOREM  KH 

where  [2=1-2,  [3=1.2.3,  [4=1.2.3-4,  etc.,  these  symbols 
being  rend  factorial  two,  factorial  three,  factorial  four,  etc. 
Dividing  sin  x  by  cos  x  and  cos  x  by  sin  x,  we  find 

III.  Uinoo-x+  ^  +l5^+  315  +      ' 

IV.  cota!  =  I-^-^-2^'-.... 

ic     3      46      945 

The  series  for  cos  x  and  sin  x  are  convergent  for  any  value 
of  X.  In  the  expansion  for  any  trigonometric  function  the 
radian  measure  of  x  must  be  used  in  the  series. 

Tables  of  the  numerical  values  of  the  trigonometric  functions 
for  any  set  of  angles  may  be  computed  by  means  of  the  series 
1,  II,  III. 

Examples,     i.    Compute  sin  10°  correct  to  four  decimals. 

/'a;  =  ^  =  0.17453\ 

2.  Compute  sin  12°.     fx  =  ^.  =  0.20du\ 

3.  Compute  cos  10°,  tan  10°. 

4.  Compute  sin  20°  from  Ex.  1  and  3. 

5.  Find  sin  80°,  cos  80°,  cos  78°,  cos  70°. 
(4)   Trigonometric  products,     sin  x  =  0  when 

x  =  0,  x=±7r,  x=±2'rr,  x=±^7r,  .... 

This  fact  suggests  that  sin  x  can  be  expressed  as  a  product 
of  factors,  and  indeed  an  infinite  number  of  factors.  Likewise, 
cos  x  =  0,  when 

_7r  _    .^"^ 5  TT 

X  —  ±—,    X—  ±  —,    X  —  ±—^,   •■-. 

Arranging  the  factors  properly, 

-^H'-(^r)l'-(lf)*H-(if)T"- 


108 


PLANE   TRIGONOMETRY 


[§70 


70.  Summation  of  Series-  In  analysis  it  sometimes  becomes 
necessary  to  sum  the  following  series : 

(1)  aS'i  =  sin  ^  +  sin  (^  +  a)  +  sin  (<9  +  2  a)  +  ... 

+  sin[^  +  (7i-l)a], 

(2)  aSj  =  cos^  +  cos(0+ a)+cos(^  +  2a)+  ••• 

+  cos  [^  -f  (w  —  1  )a] . 

iS'j  is  a  series  of  n  sines  in  which  the  angles  are  in  arithmeti- 
cal progression,  the  common  difference  being  a.  S^  is  a  similar 
series  of  cosines. 

To  find  the  sum  S^,  multiply  both  members  by  2  sin  -  • 

At 

2sin->S'j  =  2  sin  -  sin  ^+  2  sin -sin  (^  +  «)+  ••• 

U  U  Li 


+  2  sin  ^  sin  [^  +  (n  -  l)a] 

cos(^-|^-cos(^+|')}  +  |cos(0  +  |)-cos(^^  +  |»)j 


+ 


+ 


cosf  ^  +  -aj— cosf^-f^aj  [ 


+ 


cos(^H aj-cosf^H a)    , 


§  39  (22') 


=  cosr^-|j-cos(^^  +  "^^       ") 
=  2sinr^  +  (w-l)|'jsin^. 


Dividing  by  2  sin  - ,  we  have 


^1  = 


sinp  +  (w-l)| 


sin 


na 


•    a 
sin- 


§70] 


TRIGONOMETRIC   SERIES 


109 


In  a  similar  manner,  by  multiplying  S^  by  2 sin-  and  sepa- 

rating  the  double  products  as  in  S-^^  we  may  reduce  the  value 
of  aSj  to 

<Sj=cos^+cos(^  +  a)  +  cos(^  +  2«)+  •••  +  cos [^  +  (w -  l)a] 


cos 


'^+(.1-1)1" 


sin 


na 


sm 


EXAMPLES 
Verify  the  following : 

1.  sinr+sinfar+^j+sinfaj+^W  •••  +sin   2;+(w— 1)^ 

=  2  sin  x-\-(n  —  V)—   sin 

Suggestion.     Common  difference  =  ^ ,  compare  Sy 

2.  cosa;+cosf  a;— ^j  +  cosf  X—  ^)+  •••  4-cos  a;— (n  — 1)^ 

=  -2cos  a;-(w-l)|  sin^-^V 
r Suggestion.    The  common  difference  is  —  ^;  apply  ^2  with  a  =  —  ^. 


3.    sina;  + sin2a;  +  sin  3a;+  •••  +sinwa;  = 


sm(w  +  l)-sin  — 


sin 


4.    cosa;+co8  2x4-cos3a;+  •••  +coswa;  = 


(,  -i^X    .    fix 
w  +  l)^sin  — 

.     X 

sm- 

2 


5.    sin  a;  +  sin  3  a;  +  sin  5  a:  +  •••  -f  sin(2w  —  l)a:  = 


sin^wa; 


sma: 


110  PLANE   TRIGONOMETRY  [§§70-71 

cos  nx  axil  nx 


6.    cos  a; + cos  3  a;  +  cos  5  a; -I-  •••  +cos(2n— 1)2:  = 


sill  a; 

sin  2  nx 
2  sin  X 


•    o     ,     ■    A      ,     •    a     ,  .     •    o  sin(w-f-l)a:sin  wa: 

7.  sinza;+sin4a;  +  sinoa:-f-  •••  +sin2wa;= — ^= — ' — ^^ 

sin  a; 

8.  cos 2 a;  +  cos 4 a;  +  cos 6x+  ••■  +  cos 2 nx 

_  cos  («  4-  1)  a;  sin  wa; 


sin  X 


9.    siii2a;+sin2(a:4-a)4-sin2(a;4-2a)4-  •••  4-sin2[a;+(w  — l)a] 

_  w  _  cos  [2  a;  4-  (?t  —  !)«]  sin  >m 
2  2  sin  a 

[Suggestion.     Multiply  this  series  by  2  and  separate  each  term  thus : 
2  sin^a:  =  1  —  cos  2  a;,   2  sin^  (x  +  a)  =  1  -  cos(2  x  +  2  a),  ... 
2  sin2  [x  +  (n  -  l)a]  =  1  -  cos  [2  a:  +  (n  -  1)  2  a]  ; 
add  and  sum  the  cosines  as  iu  Ex.  8.] 

10.    cos^  X  +  cos^  (a;  4-  «)  4-  cos^  (a;  4-  2  a)  4-  •  •  • 

,         2r     .  ^        i\   n      w  .  COS  r2a:  4-^^  —  l)al  sin  w« 
4-  cos^  ix  +  (n  —  l)a]  =  -  4 >= !-^ — -, ^^— ' . 

2  2  sin  a 

71.    The  Exponential  Series.     If  we  take  the  Binomial  Series, 

(l-hzy  =  l  +  nz  +  -~^ — ^z2_|._v ^v 1^_^  ...^ 

and  substitute 


g  =  — ,  n  =  ma;, 


we  find 


my 


Now,  divide  the  numerator  and  denominator  of  the  respective 
fractions  by  m,  m\  m^  •■•,  and  finally  let  m  =  ao  ;  then  we  have 

.r^      3^      3^ 

a  n  \± 


'tl(i+^'"^i+^+':^+'^+'-+ 


limit 
m 


§§71-72]     EXPONENTIAL  AND   HYPERBOLIC   FUNCTIONS     111 
The  numerical  value  of  this  series,  when  2;  =  1,  is  denoted  by  e. 

(^)  e  =  1  +  1  +  i  +  1  +  -t  +       =  2.7182818  ••• 

|_2      id      [^ 

{B)  e^  =  \  +  x  +  ^  +  ^  +  ^  +  ...=  (2.7182818  ...)* 

[2.    L*.    L£ 

Series  (jB)  is  called  the  exponejitial  series;  e  is  the  base  of 
the  Napierian  Logarithmic  %ystem. 

72.  Euler's  Formulas.  Series  (5)  readily  identifies  the  ex- 
ponential functions  with  the  trigonometric  functions.  In  (^) 
substitute 

X  =  i0,  where  z  =  V—  1,  i"^  =  —  1,  ...,  §  65 

|2^[4  ^{       [3^15 

=  cos<9^-^sin^,  §69,1,11. 

Changing  i  to  —  z, 

g-'^  =  cos  ^  —  i  sin  ^. 

Subtracting  and  dividing  by  2  z,  and  adding  and  dividing  by 
2,  respectively,  we  have  Euler's  Formulas  for  sin  ^  and  cos^. 
tan  0  =  sin  0  -i-  cos  0. 


2i       • 

VI. 

COS  9  = 

gie  +  g-ie 

• 

2 

k^I. 

tan  6  = 

The  reciprocals  of  these  fractions  define  the  esc  0,  sec  0,  cot  0, 
respectively. 

These  analytic  definitions  of  sin  0,  cos  0,  tan  0,  may  be  em- 
ployed instead  of  the  ratio  definitions  given  in  §  2. 


112  PLANE   TRIGONOMETRY  [§§72-73 

EXERCISES 

Prove  the  following  identities  by  use  of  Euler's  definitions, 
V,  VI,  VII. 

1.  sin  2^=2  sin  0  cos  0. 

2.  COS  26  =  cos^6  —  sin^^. 

3.  sin  (a;  +  2/)  =  sin  re  COS  «/  + COS  a;  sin  y. 

4.  sin X  +  sin  y  =  2 sin     'Z^ cos  — — ^ • 

^  2  2 

5.  sin^a;  +  cos^a:  =  1. 

6.  sec^a;  —  tan^aj  =  1. 

7.  Write  trigonometric  values  for  each  of  the  following : 

(1)    e~';     (2)   €-;     (3)    g^?'^;     (4)    e'~* ;     (5)    g"*' ;     (6)    e~''' ; 

IT 

(7)   e~"';    (8)  e*(-+«>. 

8.  Express  in  exponential  notation  :   (1)  cos  30° +  ^  sin  30°; 

^^^    Vl"'        ^  ^  2 '        ^^^  -2-  '        ^^^   -^  ' 

(6)  yi-^^^" 


2 

9.    Prove  the  following:    (1)  e'(«+2,r)^  ^ta.   (^2)  e2+t:r^_g2. 
(3)  e^'^  =  ie^;   (4)  e2+2.,  =  g2, 

10.   Find   approximately  the  value  of    Ve   by  substituting 

a:  =1  in  (5),  §  71. 

73.    The  Hyperbolic   Functions.     In  V,   VI,  VII,   §  72,   let 
6  =  ix, 

sinia;  =  — — —  =  ^ ,    cosea;  = , 

e~^  —  e^         .e^  —  e~^ 
tan  IX  = — —  —i- 


i{e  •*■+  e^)        e^  -^  e  ^ 


§  73]         EXPONENTIAL  AND   HYPERBOLIC   FUNCTIONS        113 

^  p^  p      ^  P'^'    ..1  ■   p~^  p^   p      ^ 

The  fractions   — ,   ■ -^ — ,   are  taken  as  defi- 

2  2  e'^  +  e"-^ 

nitions  of  the  hyperbolic  sine  of  x^  hyperbolic  cosine  of  x,  and 
hyperbolic  tangent  of  x,  respectively.  These  functions  are 
written  : 


VIII. 

IX. 

C08ll£C=^^^±^. 

X. 

tank  x-S"'**^ -«"-«"' 

cosh  X     e*  +  c"* 

(1)  Relations  betiveen  trigonometric  and  hyperbolic  functions. 
From  the  above  definitions  it  is  seen  that 

sin  ix  =  i  sinh  x,  cot  ix  =  —  i  coth  x, 

cosix  =  cosh  X,  sec  ix  =  sech  x, 

tan  ix  =  i  tanh  x,  esc  «a;  =  —i  csch  a;. 

This  table  of  relations  enables  us  to  determine  the  identities 
existing  among  the  hyperbolic  functions,  corresponding  to  the 
trigonometric  identities. 

(2)  Identities.  Let  the  student  verify  the  following  identi- 
ties among  the  hyperbolic  functions  : 

CI)  cosh^  X  —  sinh'^  a;  =  1.              ^/>v    ,      i  o            2  tanh  x 
^  ^  (6)  tanh 2x  =  — — —  • 

(2)  sech2a:  +  tanh2  2;  =  l.  l  +  tanh^a; 

(3)  coth2  ^  _  csch2  ^  ^  1^  (7)  sinh  (  -  a:)  =  -  sinh  x. 

(4)  sinh  2  a;  =  2  sinh  x  cosh  x.  (8)  cosh  (  -  a;)  =  cosh  x. 

(5)  cosh  2  a:  =  cosh^  x  +  sinh^  x. 

(9)  sinh  (x  +  y')=  sinh  x  cosh  ^/  ±  cosh  x  sinh  y. 

(10)  cosh  (x  ±y')=.  cosh  a;  cosh  y  ±  sinh  x  sinh  y. 

^ii\  4.     1  ^     ,     \       tanh  a:  ±  tanh  V 

(11)  tanh(a:±y)=- :; r^- • 

^     ^  V       ^^     1  ±  tanh  a;  tanh  y 

(12)  e^  =  cosh  a;  +  sinh  x. 

(13)  sinh  a;  +  sinh  y  =  2  sinh  ^  ":  ^  cosh  ^~^ ■ 

(14)  cosh  X  —  cosh  y  =  2  sinh  ^"t  '^^  sinh  — — ^• 

4U  ^ 


114 


PLANE   TRIGONOMETRY 


[§73 


(3)  Infinite  series  forms  for  sink  x,  cosh  x.     Take  the  series 
for  sin  a;,  cos  a;,  tanx,  II,  I,  III,  §  69, 

sin.=  .-|+|-.., 
C0S2;  =  1  — — +  — -  •••, 

t^nx  =X  +  —+^-:r+    •-, 

o        lo 

and  replace  x  by  ix,  giving 

sinhic  =  a5  +  ^+^+  ..-, 

II       |0- 


12.     I£ 

tanh x  =  X  —^  J-  ^^^ 
3        15 


■■ 

1 II 1 1 1 

|Y- 

I 

::;:;;;;!- 

il 

:::::::::!  1  j! 

ni-. 

. . 

;;;:;i;;;:i^;;; 

1  '■'■ 

in 

7 

/ 

9 

^ 

i 

i 

1 

|: 

liiilT 

Fig.  51. 


(4)  Graphs  of  the  hyper- 
bolic functions.  The  numeri- 
cal values  of  the  hyperbolic 
functions  for  various  values 
of  the  variable  x  are  col- 
lected in  Tables  of  Hyper- 
bolic Functions.  The  ap- 
proximate values  of  sinh  a;, 
cosh  x,  tanh  x  are  shown  upon 
the  following  graphs.  Fig. 
51,  where  the  ordinates 
parallel  to  the  !F-axis  are  the 
values  of  these  functions  for 
the  corresponding  values  of 
X.  The  graphs  of  the  cscli  a:, 
sech  a;,  and  coth  x  are  not 
shown,  but  these  may  be 
easily  constructed  upon  the 
same  diagram  by  taking  re- 


ciprocals of  the  ordinates   of   the    sinh  a;,  cosh  x,  and   tanh  x, 
respectively. 


^7:'.]  KXPONKNTIAL    AND    HYPERBOLIC    FUNCTIONS       115 

(5)    The  inverse  h i/perhoUc  functions.      If  we  write 

.    ,           e'-e~' 
y  =  smli  X  = , 

we  may  express  z  in  terms  of  y  by  an  inverse  notation  similar 
to  that  employed  in  inverse  trigonometric  notation. 

i/  =  sinh  a\     x  =  sinh"^  y. 

By  means  of  the  exponential  value  of  sinh  a:,  we  may  obtain 
another  expression  for  x  in  terms  of  y.     Thus, 

e-^  —  e'-" 

=  «/.     e-'-  -1  =  2  ye\ 

or  (?-''  —  -  ye''  —1  =  0,  a  quadratic  in  e^. 

Solving,  t;r  =  y  ±  -\' y'-  +  1, 

the  negative  sign  being  excluded  since  e^  is  positive.     Take 
logg  of  each  memljer. 


^  =  ifjgAi/  +  ^y"  +  1)  =  ^iiiii"^  y- 

EXERCISES 
Prove  the  followins: : 


1.  cosh-i  x  =  log,(.r  +  ^  x'^  —  1). 

2.  tanlr^  x  =  -l  loc^j  — ^■^- 

"      "  \1  —  J 


1  1  +  A  1  -  .r2 

3.    seclr^  X  =  cosh  ^-  =  log„  — 


4.    sinh~i  a:  =  cosh'^Vl  +  a^  =  tunh   i 


r 

X 


Vl+a;^ 


5.  tanlri  _^._^  tanh-i  y  =  tanh  ^  / ^• 

1  +  xy 

6.  cosh  0  =  1,    cosli  —  =  '),  sinh  iri  =  0. 

7.  sinh  —  =  /.   cosh  tti  =  —  1.  tanh  0  =  0. 

2 

8.  sinh  2  mri  =  0,  cosh  2  liiri  =  1.  tanh  mri  =  0. 


116  PLANE   TRIGONOMETRY  [§74 

74.    The  Gudermannian.     If  an  angle  d  is  related  to  a  num- 
ber X,  so  that 

sec  0  =  cosh  X, 
then  0  is  defined  by 

0  =  gudermannian  of  x, 

or  briefly,  ^  =  gd  a;  =  sec~^(cosh  x'). 

Inversely, 

X  =  inverse  gudermannian  of  0, 

or  a;  =  gd~i  ^  =  cosh"  1  (sec  ^). 

When  sec  0  =  cosh  x, 

the  following  relations  are  true  : 

cos  0  =  sech  X,  sin  0  =  tanh  x, 

tan  0  =  sinh  x,  esc  0  =  coth  x, 

cot  ^  =  csch  X,  i.      ^      4.     u  ^ 

tan- =  tanh-- 

2  2 

Any  one  of  these  relations  defines  0  as  the  gudermannian  of  x. 


PART   II 

SPHERICAL   TRIGONOMETRY 


CHAPTER  X 

GENERAL   DEFINITIONS.     THE    RIGHT  TRIANGLE 

75.  Definitions  and  Geometric  Properties.  From  spherical 
geometry  we  recall  the  following  facts : 

(1)  The  intersection  of  the  surface  of  a  sphere  by  a  plane 
through  its  centre  is  a  great  circle. 

(2)  A  great  circle  divides  the  surface  of  a  sphere  into  two 
equal  parts  called  hemispheres. 

(3)  Two  great  circles  upon  the  same  sphere  divide  its  sur- 
face into  four  parts  each  of  which  is  called  a  Lune.  In  Fig. 
52,  BCB'AB  is  a  lune,  bounded  by  the  semi-circumferences 
BOB'  and  BAB'. 

(4)  The  angles  of  a  lune  are  equal,  and  are  measured  by  the 
diedral  angle  between  the  planes  of  the  great  circles  forming 
the  sides  of  the  lune.  In  the 
drawing,  angle  B  is  meas- 
ured by  the  diedral  angle 
CBB'A.  Since  tangents  to 
arcs  BC,  BA  at  B  are  per- 
pendicular to  the  edge  of  the 
diedral  angle,  the  spherical 
angle  B  is  measured  by  the 
plane  angle  between  the  tan- 
gents BU  Bt'  to  the  arcs  BC 
and  BA  at  B. 

(5)  Any  third  great  cir- 
cle will  divide  a  lune  into 
two  parts  called  spherical 
triangles.  Thus,  arc  AQ, 
Fig,  52,  divides  the  lune  into  the  spherical  triangles  ABO  and 
AB'C. 

117 


118 


SPHERICAL   TRIGONOMETRY 


[§§  75-76 


(6)  0  is  the  apex  of  a  spherical  pyramid  whose  base  is  the 
spherical  triangle  ABC  The  face  angles  of  this  pyramid  are 
equal  to  the  corresponding  sides  (measured  in  degrees)  of  the 
spherical  triangle ;  in  Fig.  52,  a  =  Z  ^  0  (7,  5  =  Z  (7  OA,  c=ZA  OB. 
The  diedral  angles  of  the  pyramid  are  equal  to  the  correspond- 
ing angles  A,  B,  C,  of  the  spherical  triangle. 

(7)  The  length  of  a  side  a  of  a  spherical  triangle  expressed 
in  linear  units  is  given  by  arc  a  =  radius  x  radian  measure  of  a. 
The  length  of  the  radius  of  the  sphere  is  usually  not  considered 
in  the  trigonometric  discussion  of  a  spherical  triangle. 

(8)  A  side  of  a  spherical  triangle  lies  between  0°  and  180°; 
likewise  an  angle  lies  between  0°  and  180°.  The  following 
limitations  should  be  recognized  (see  notation,  Fig.  52): 

(1)  0°<a  +  b  +  c<  360°, 

(2)  180°  <A  +  B  +  C<  540°. 

(9)  The  angles  and  sides  of  a  spherical  triangle  are  in  the 
same  order  of  magnitude  ;  if  a  >  6 > <?,  then  will  A>  B>  C. 

76.  The  Polar  Triangle.  To  any  spherical  triangle  ABC 
there  corresponds  another  called  its  polar  which  may  be  con- 
structed as  follows : 
take  each  vertex  A, 
B,  C,  as  a  pole  and 
describe  arcs  (90° 
away)  B'C,  C'A\ 
A'B',  forming  the 
spherical  triangle 
A'B'C,  Fig.  53. 
The  triangle  A'^' (7' 
is  the  polar  of  the 
triangle  ABC 

In  geometry  it  is 

shown  that  the  sides 

and     angles     of    a 

spherical      triangle 

ABC  and  the  angles  and  sides  of  its  polar  A'-B'O"  are  related 

by  the  following  equations  : 

(1)  J  =  180°  -a',  B^  180°-  6',  C  =  180°  -  c', 

(2).  a  =  180°  -A',b=^  180°  -B',c^  180°  -  C 


Fig.  63. 


§§  76-78:" 


THE   RIGHT   TRIANGLE 


119 


Theorem.  Tlie  angles  of  a  spherical  triangle  are  the  supple- 
ments of  the  corresponding  sides  of  its  polar;  the  sides  of  the 
given  triangle  are  the  supplements  of  the  corresponding  angles  of 
the  polar  triangle. 

Polar  triangles  are  so  related  that  the  vertices  A,  5,  C,  of 
one  are  poles  of  the  corresponding  sides  a',  h\  c'  of  the  other, 
and  the  vertices  A\  B\  O'  are  poles  of  a,  J,  c. 

THE   RIGHT   SPHERICAL   TRIANGLE 

77.  Definitions.  If  one  angle  (7,  Fig.  54,  be  90°,  the  triangle 
ABO  is  called  a  right  spherical  triangle.  The  side  c  opposite 
tlie  right  angle  is  called  the  hypotenuse. 

(1)  The  angles  A,  B,  may  both  be  acute,  both  obtuse,  or  one 
acute  and  the  other  obtuse. 

(2)  The  sides  a,  b,  lie  in  the  same  quadrant  as  the  opposite 
angles.  If  ^  >  90°,  a>90°;  if  5>90°,  b>90°;  if  ^<90°, 
a<90;  if  jB<90°,  6<90°. 

(3)  The  hypotenuse  c  is  smaller  than  90°  if  A  and  B  are 
both  smaller  or  both  larger  than  90° ;  e  is  larger  than  90°  if 
A  is  smaller  than  90°  and  B  larger  than  90°,  or  inversely. 
See  §  81. 


78.  Trigonometric  Relations.  In  the  spherical  trisLUgle  AB 0, 
Fig.  54,  let  0  =  90°,  and  let  A.  B,  be  acute.  To  measure  the 
angle  A  draw  at  any 
point  A'  in  the  edge 
of  the  diedral  angle 
CAOB,  a  plane  per- 
pendicular to  OA ;  the 
traces  of  this  plane 
upon  planes  OAC  and 
OAB  are  the  lines 
A'C\  A'B\  and  the 
angle    BA'  C    equals 

the   angle  A.      Draw  Fi<T.  54. 

B'C.     The  following 
are  plane  right-angled  triangles :  A' C B' ,  OA' C\  OA'B',  00' B'. 


120  SPHERICAL   TRIGONOMETRY  [§§78-79 

Defining  sin  A,  cos  A^  and  tan  A  from  the  drawing,  we  have 

.     ,  _  B'C  _  B'C      OB'  _  sin  a 
^'"     "  AB'  ~  A'B'  ""  OB'  ~  '^c' 

A_A'G'  _A'C'      OA'_tsinb 
''''^  ^  ~  A'B' '  A'B'  ""OA^'  t^' 

A^^'O'  ^B'C      00'  ^tan  a 
*""         A'C     A'O'      00'      sin  6' 

Also,  from  triangle  B'  OA'^ 

j,,^.,  OA'      OA'     00'  r 

cosB' 0A'  =  cose  =  -^=-^x-^^==  cos  a  cosb. 

By  drawing  perpendiculars  to  the  edge  OB  and  constructing 
a  measure  of  angle  B,  we  could  derive  similar  results  for  the 
functions  of  B,  namely  : 

T>      sin  b  T>      tan  a    ^       t>      tan  b 

sinij=  - — ■,  cos  B  = -,  tanij  = 

sin  e  tan  c  sin  a 

Taking  the  reciprocals  of  tan  A  and  tan  B,  and  multiplying 

their  values  together, 

I  A       i.  T>      sin  b      sin  a  , 

cot  A  cot  B  = X r  =  cos  a  cos  6. 

tan  a      tan  b 

Hence,  cos  e  =  cos  a  cos  b  =  cot  A  cot  B. 

Again, 

A      tan  b      sin  b      cos  e     sin  b      cos  a  cos  5      sin  b 

cos  J.  = = X = X = X  cos  a 

tan  <?     coso      sine      cos 6  sine*  sine 

=  sin  B  cos  a. 
Similarly,  cos  B  =  sin  A  cos  b. 

79.  Important  Formulas.  Collecting  the  results  of  §  78,  we 
have  the  following  formulas  relating  to  the  right  spherical 
triangle  : 

(1)  sin  A  =  ^5«,  (5)  sin  B  =  ^, 

sine  sine 

(2)  cos^  =  ^,  (6)  cosB=*^, 

tan  c  tan  c 

(3)  tan^  =  ^,  (7)  tanB=*^, 

sin  6  sin  a 

(4)  cos  ^  =  sin  i;  cos  a,  (8)  cos  J5  =  sin  A  cos  6, 

(9)  cos  c  =  cos  a  cos  6  =  cot  4  cot  B. 


§§  79-80]  NAPIER'S   RULES  121 

These  formulas  are  true  for  any  right  spherical  triangle, 
whether  the  angles  and  sides  be  acute  or  obtuse. 

It  should  be  noticed  that  the  values  of  sine,  cosine,  and 
tangent,  formulas  (1),  (2),  (8),  (5),  (G),  (7),  are  very  similar 
to  the  definitions  of  those  functions  in  a  plane  triangle.  This 
similarity  enables  one  to  remember  the  trigonometric  relations 
of  the  angles  and  sides  of  the  right  spherical  triangle. 

A  right  spherical  triangle  may  he  completely  solved  for  all  its 
parts  tvhen  any  two  parts,  other  than  the  right  angle,  are  known. 

80.    Napier's  Rules  of  Circular  Parts.     Another  method   of 
remembering  the  formulas  (1)  to  (9)  of  §  79  is  embraced  in 
what  are  called  Napier'' s  Rules  of  Circular 
Parts. 

Let  a  right  spherical  triangle  be  given 
with  the  usual  notation.  Fig.  54,  the  right 
angle  being  C.  Then  write  a,  h,  comple- 
ment of  A,  complement  of  c,  complement 
of  B,  Fig.  55.  The  notation  shown  on 
the  drawing  consists  of  five  parts,  known 
as  Napier's  Circular  Parts. 

If  any  part  as  eo.  cbe  taken  as  a  middle 

1  1-1  ,        CO.  A 

part,  the  next  two  parts  to  the  right  and 

left,  CO.  A,  CO.  B  are  called  the  adjacent  parts., 

and  the  remaining  two  parts,  a,  h,  are  called  the  opposite  parts. 

Napier's  Rules.  (1)  The  sine  of  the  middle  part  is  equal 
to  the  product  of  the  tangents  of  the  adjaceiit  parts. 

sin(jnid.  pt.^=  tan(^adj.  pt.^  x  tan(adj. pt.^. 
(2)    The  sine  of  the  middle  part  is  equal  to  the  product  of  the 
cosines  of  the  opposite  parts. 

sin(mid.  pf.^=  cos(pp.  pt.')  x  cos(^op.  pt.^. 

As  illustration  of  these  rules,  identify  each  of  the  following 
with  some  one  of  the  formulas  (1)  to  (9),  §  79  : 

1.  sin(c(?.  c)  =  cos  c  = 

2.  sin(a)  = 

3.  sin  {co.  B}  =  cos  B  = 

4.  sin(^>)  = 

5.  sini^co.  A)=  cosA  = 


122  SPHERICAL   TRIGONOMETRY  [§§  81-82 

81.    Relative  Dimensions  of   Sides  and  Angles.     (1)  Given 
•  a  <  90°,  6  <  90°.     In  this  case, 

(+)  +         + 

cos  0  =  COS  a  cos  b 

shows  cos  e  positive  ;  hence,  c<  90°. 

.,                           <+>    .      tan  5        ''+>o      tan<? 
Also,  cosA  = ,    cosij  = 

tan  c  tan  c 

show  cos  A  and  cos  B  positive  ;  hence  A  <  90°,  B  <  90°. 

(2)  Given  a  >  90°,  b  <  90°.     Formula  (9),  §  79, 

(-;  -  + 

COS  c  =  cos  a  CO.S  o, 

shows   cos  e   negative ;  hence,  c  >  90°,  th6  supplement  of  the 
tabular  angle  determined  by  formula  (9). 
Formulas  (2),  (6),  §  79,' 

+  — 

^"^    .      tan 5       "•"    r»     tana 
008^  =  —; — -,    cosij  =  -3 — , 

tan  c  tan  o 

show  A  >  90°,  B  <  90°. 

(3)  Given  a  >  90°,  b  >  90°.     Formula  (9), ' 

(+) 

cos  c  =  COS  a  cos  J, 

determines  cos  c  positive  ;  hence  c  <  90°. 

A ,                         ^~^    A      tan  a      ^~^    t>      tan  b 
Also,  cos  A  = ,    cos  B  = 

tan  c  tan  c 

show  that  cos  A^  cos  ^  are  negative  ;  hence,  A  >  90°,  B  >  90°. 

Theorem.  In  any  right  spherical  triangle  the  hypotenuse  c  is 
acute  if  a  and  b  lie  in  the  same  quadrant ;  c  is  obtuse  if  a  and  b 
lie  in  opposite  quadrants. 

82.  The  Isosceles  and  Quadrantal  Triangles.  If  two  sides  of 
a  spherical  triangle  be  equal,  the  opposite  angles  are  equal,  and 


§§  82-83] 


THE   RIGHT   TRIANGLE 


123 


the  triangle  is  isosceles.     An  isosceles  spherical  triangle  may 

be  divided  into  two  symmetrical  right  triangles,  ADB^  CDB^ 

by  drawing  the  arc  of  a  great   circle 

from  5,   Fig.  b^.,  perpendicular   upon 

side   h.     This   perpendicular  'p  bisects 

the  angle  B  and  the  opposite  side  h. 

The    solution   of   an   isosceles  triangle 

ABO  depends  upon  the  solution  of  the 

two  right  triangles  ADB,  CDB. 

If  a  side  be  90°,  the  triangle  is  called 
a  quadrantal  spherical  triangle.  The 
solution  of  a  quadrantal  triangle  may 
be  made  to  depend  upon  the  solution 
of  a  right  spherical  triangle  by  tak- 
ing the  polar  triangle,  one  of  whose 
angles  will  be  90°,  the  supplement  of  the  given  side.     See  §  76. 

To  solve  a  quadrantal  triangle,  solve  its  polar  and  take  the  sup- 
plement of  each  part  of  the  polar. 

83.    Solution  of  Right  Spherical  Triangles.     Six  cases  arise  in 
the  solution  of  right  spherical  triangles. 

Case  I.     Given  two  sides,  a,  h. 

Case  II.     Given  one  side  and  the  hypotenuse,  a,  c. 

Case  III.     Given  two  angles,  A,  B. 

Case  IV.     Given  one  angle  and  the  adjacent  side,  A,  h. 

Case  V.     Given   one   angle   and   the   opposite  side,  A^  a. 
Double  solution. 

Case  VI.     Given  one  angle  and  the  hypotenuse,  A.,  c. 

That  Case  V  has  a  double  solution  may  be  seen  from  the 
drawing.  Fig.  57. 

B 


Fi-.  57. 


124         '  SPHERICAL   TRIGONOMETRY  [§  83 

In  calculating  the  unknown  parts  of  a  triangle  falling  under 
any  one  of  the  above  Cases,  (1)  either  select  a  proper  set  of 
formulas,  §  79,  or  employ  Napier's  Rules  to  obtain  the  required 
trigonometric  relations ;  (2)  use  logarithms  in  performing  the 
operations  of  multiplication  and  division. 

EXERCISES 

Solve  the  following  right  spherical  triangles  for  each  un- 
known part: 

1.  a  =64°  20',   5  =  70°  24'. 

2.  a  =  49°  28',   c  =  65°25'. 

3.  5  =  100°  10',   ^  =  38°  47'. 

4.  a  =  120°  30'  10",   B  =  98°  27'  10". 

5.  c  =  110°  30' 10",   ^=46°  21' 30". 

6.  ^  =  41°  48',   5  =80°  30'. 

7.  6  =  140°  28',   c=110°29'. 

8.  6  =72°  20' 10",   i?  =  80°  10' 20". 

9.  a  =130°  20',   i  =  105°48'. 

10.  A  =  98°  25',   5  =  104°  10'. 

11.  Find  the  perimeter  of  the  right  triangle  in  which  a  =  48°, 
h  =  60°,  the  radius  of  the  sphere  being  15  in. 

12.  Find  the  area  of  the  right  spherical  triangle  determined 
by  a  =  36°,  A  =  36°,  the  radius  of  the  sphere  being  30  in. 

Note.     The  surface  of  a  sphere  =  4  ttR^. 

13.  If  a  spherical  triangle  have  two  right  angles,  show  that 
the  opposite  sides  are  quadrants,  and  explain  a  method  for 
finding  its  area  when  the  radius  is  known. 

14.  Solve  the  isosceles  triangles: 

(1)  vi  =  5  =  54°28',    (7=68°. 

(2)  A  =  75°  34',    a  =  38°  28',    h  =  c. 

(3)  ^=98°  24',   a  =  c=124°. 

15.  Solve  the  quadrantal  triangles  : 

(1)  a  =  90°,    5=48°,    (7=38°. 

(2)  A  =  100°,   c?  =  98°,   5  =  90°. 


§83]  THE   RIGHT  TRIANGLE  '  125 

16.  A  vessel  sails  directly  east  from  Sandy  Hook  (latitude, 
40°  28'  N.;  longitude,  74°  1'  VV.)  and  continues  upon  the  arc 
of  a  great  circle.  Find  the  latitude  at  which  it  crosses  the 
meridian  of  65°  W.  What  will  be  the  course  of  the  vessel 
when  crossing  that  meridian  ? 

Note.     See  §  91  for  definitions. 

17.  If  a  vessel  sails  directly  west  along  an  arc  of  a  great 
circle  from  San  Francisco  (latitude,  37°  48'  N.;  longitude,  122° 
28'  W.),  what  distance  will  it  have  sailed  and  what  direction 
will  be  its  course  on  arriving  at  the  180th  meridian  ? 

Note.     Take  the  radius  of  the  earth  as  3956  mi. 

18.  A  ship  sails  due  east  from  Boston  (latitude,  42°  21'  N. ; 
longitude,  71°  4'  W.)  upon  the  arc  of  a  great  circle  at  the  rate 
of  16  knots  per  hour.  Find  the  latitude  and  longitude  of  the 
ship  after  48  hours'  sailing. 

Note.  1  knot  =  1  nautical  mile  =  1  geographical  mile  =  1'  arc  of  a 
great  circle  upon  the  earth. 

19.  A  triangular  p3'ramid  0-ABC  has  the  diedral  angle 
along  the  edge  00  =  90°,  and  the  face  angles  AOC=  48°,  and 
BOC—  69°.  Find  the  face  angle  BOA^  and  the  diedral  angles 
along  the  edges  OA  and  OB. 

20.  If  the  three  edges  of  a  cube  through  a  point  0  are  OA^ 
OB.,  and  OC,  find  the  diedral  angle  made  by  the  plane  ABC 
with  either  face  of  the  cube. 


CHAPTER   XI 


THE   OBLIQUE    SPHERICAL   TRIANGLE    ^ 

84.    The  Theorem  of  Sines.     Let  a  spherical  triangle  be  repre- 
sented by  ABC,  Fig.  58.     If  no  angle  be  a  right  angle,  the 

triangle  is  called  oblique. 

Draw  through  angle  B  the  arc  of 
a  great  circle  to  meet  the  side  h  at 
right  angles  in  D.  Then  ABB,  CBB 
are  right  spherical  triangles.     From 

§  ^9, 

.       sin  p      .     .v     sin  p 

sinA  = ^,  sm  6=    .    ^  - 

sin  c  sin  a 

Dividing,  we  have, 

sin  A      sin  a 


sin  C 


sin  c 


By  drawing  an  arc  through  A  perpendicular  to  a,  we  may 
derive  similarly, 

sin  C  _  sin  c 
sin  B     sin  h 

Rearranging  these  two  formulas, 
sin  A      sin  B 


(1) 


sin  a 


sin  6 


sin  C 


Slll<? 


Theorem  of  Sines.  In  any  spherical  triangle  the  sines  of  the 
angles  are  proportional  to  the  sines  of  the  opposite  sides. 

Observe  the  similarity  to  the  corresponding  theorem  in  plane 
trigonometry. 

The  theorem  of  sines  enables  us  to  compute  a  fourth  part, 
when  two  angles  and  a  side  opposite  one  of  the  angles  are 
known,  or  when  two  sides  and  an  angle  opposite  one  of  the 
given  sides  are  known. 

126 


§  85]  THE   OBLIQUE   SPHERICAL   TRIANGLE  127 

85.    The  Theorem  of  Cosines.      Sides  and  One  Angle.      The 

angle  B  in  the  triangle 
ABC,  Fig.  59,  is  meas- 
ured by  the  plane  angle 
between  tlie  tangents 
to  AB  and  CB  at  B. 
Let  these  tangents  in- 
tersect the  radii  OA 
and  00  produced  in  L 
and  M. 


d'j. 


Then  from  the  plane  triangles  L  OM  and  LBM,  we  have 
LM^  =  TO^  +  M^ -2L0x  MO  x  cos  6, 
LM-  =  LB^  -h  MB^  -2LBx  MB  x  cos  B 

Subtracting, 


or, 


0  =  LO^ -  LB^  +  MO^ -  MB^ -2L0xM0xcosb 

■\-2LBxMBxcohB, 

0=0^  +  0J^-2L0xM0xcosb 

+  2LBx  MB  X  cos  B. 

Transposing  the  negative  term  and  dividing  by  2  LO  x  MO, 


we  find. 


cos  J  = 


O:^       ,  LB     MB  J. 

LOx-Md'-Ld'^Md'''''^^ 


and  substituting  the  ratios  from  the  right  triangles  LB  0,  MB  0, 
(^2)  cos6  =  co8ccosa  +  sinc8inacosB. 

Making  symmetrical  interchanges  of  the  letters, 


(3) 


cos  c  =  COS  a  cos  6  +  sin  a  sin  b  cos  C. 
cos  a  =  cos  6  cos  c  +  sin  b  sin  c  cos  A, 


Theorem  of  Cosines.  In  any  spherical  triangle,  the  cosine 
of  any  side  equals  the  product  of  the  cosines  of  the  other  sides  in- 
creased by  the  product  of  their  sines  multiplied  by  the  cosine  of  the 
included  angle. 

Formulas  (2),  (3),  or  (4)  will  determine  a  side  when  the 
other  two  sides  and  their  included  angle  are  given;  or  they  will 
determine  the  angles  when  the  three  sides  are  given.  These 
formulas  are  not  adapted  to  logarithmic  calculation. 


128  SPHERICAL   TRIGONOMETRY  [§§  85-86 

Modified  Formula.  If  one  side  only  be  required,  with 
two  sides  and  the  included  angle  given,  it  is  sometimes  con- 
venient to  modify  formulas  (2),  (3),  (4)  so  as  to  permit  loga- 
rithmic calculation. 

For  example,  let  h,  c,  A  be  given,  then  a  is  determined  by  (4), 

cos  a  =  cos  b  cos  e  +  sin  b  sin  e  cos  A. 
Let 

tane^  .  ^f  ^   .-^^-7?  (A) 

sin  b  cos  A     cos  A 

,  1  •  .    /)  cos  b 

then,  sm  0  = 


(cos^6  +  sin^  6  cos^ -4)^ 
With  this  substitution,  cos  a  may  be  written 

cos  a  =  5*^  since +c).  (B) 

sin  6 

To  find  side  a,  first  find  6  by  equation  (^),  then  substitute 
6  in  (5)  and  determine  a.     Use  logarithms. 

86.    The  Theorem   of  Cosines.      Angles  and  One  Side.      By 

substituting  in 

cos  b  =  cos  e  cos  a  +  sin  e  sin  a  cos  B,  §  85  (2) 

the  values 

b  =  180°-B',  c  =  180°-  C",  a  =  180°-^',  5  =  180° -5', 

where  B\  C,  A',  b'  belong  to  the  polar  triangle,  we  find 

cos  (180°  -  B')  =  cos  (180°  -  0'}  cos  (180°  -  A'} 

+  sin  (180°  -  0")  sin  (180°  -A'-)  cos  (180°  -b'); 

reducing,  and  changing  signs, 

cos  B'  =  —  cos  C  cos  A'  +  sin  C  sin  A'  cos  b'. 

Omitting  accents,  we  have 

(5)  cos  B  =  —  cos  C  cos  ^  +  sin  C  sin  A  cos  b. 
Formulas  (3)  and  (4),  §  85,  by  similar  substitutions  become 

(6)  co^C  =  —  cos  ^  cos  B  +  smA  sin  B  cos  c. 

(7)  cos  ^  =  -  cos  B  cos  C  +  sin  B  sin  C  cos  a, 


§  87]  THE   OBLIQUE   SPHERICAL   TRIANGLE       ,  129 

87.  The  Half -angle  Formulas.  By  transformations  similar 
to  those  employed  in  §  50,  formulas  (2),  (3),  (4),  §  85,  and 
(5),  (6),  (7),  §  86,  may  be  changed  so  that  logarithmic  com- 
putation may  be  used  to  determine  the  angles  of  a  spherical 
triangle  in  terms  of  the  sides,  or  the  sides  in  terms  of  the 
angles. 

(1)   To  derive  tan  \B^  tan  \G^  tan  \A. 

Taking  (2),  §  85, 

cos  h  =  cos  c  cos  a  4-  sin  c  sin  a  cos  B, 
and  solving  for  cos  B,  we  find 

cos  b  —  cos  c  cos  a 


(A)  cos  B  = 


sin  c  sin  a 


Subtracting  both  members  of  equation  (J.)  from  unity,  and 

recalling  that 

l-cos^=2  sin2ij?,  §37 

we  have 

cos  c  cos  a  +  sin  e  sin  a  —  cos  b 


2  sin2 1  B  = 


sin  c  sin  a 
_  cos  (c  —  a)  —  cos  b 


sm  e  sin  a 


§34 

_  2  sin  I-  (b  -\-  c  —  a')  sin  t|  (^a-\-b  —  c)        -  og  ^22"^ 
sin  c  sill  a 
Dividing  by  2,  substituting 

a  4-  5  +  c  =  2  s,  5  +  c  —  «  =  2(«  —  a),  a  +  5  —  c  =  2(8  —  c), 
and  extracting  the  square  root, 

rr>\  •     ^   D      ^ /sin(8  — «)sin(s  — <?) 

(5)  sin  ^  B=  \l ^^ — .  -. — ^^ • 

^  sin  0  sin  a 

Adding  unity  to  both  members  of  equation  (J.),  we  have 

1   ,  Ti      n       9  1  T>      cos  b  —  Ccos  c  cos  a  —  sin  c  sin  a) 

1  -f  cos  B  =Z  COS^  I  B=  ^^^ : : 

sin  c  sin  a 

_  cos  b  —  cos  (g  -t-  g) 

sin  (?  sin  a 
_  2  sin  |-  (g  -t-  5  -f  c)  sin  ^  (g  -i-  g  —  5) 
sin  c  sin  a 

Hence,  

rn^  1   o      *  /sin  «  si"  (s  —  J) 

^  ^      sm  c  sin  g 


130  SPHERICAL   TRIGONOMETRY  [§  87 

Dividing  (5)  by  (  C),  and  writing  symmetric  results  for  the 
angles  G  and  A^  we  find  : 


(8)  tan  -  ^  =  -^^P  (^  -  <»)  sin  (a  -  c) 1^ 


sin  s  sin  (s  —  6)  sin  (s  -  6)  * 


(9)  tan^  C  =  a/^'"^^ ~ ^^ 8i»(^  -«)  = ?? , 

2  \      sin  s  sin  (8  —  c)  sin  (s  -  c) 

(10)  XfiXi\A=^^^^^^^^^^  = ^ , 

2  '      sins8m(s  — a)         8in(s  — a 

where  ^  is  defined  by 

T  _    /sin  (g  —  g)  sin  {%  —  5)  sin  (s  —  <?) 
^  sin  s 

Formulas  (8),  (9),  (10)  are  well  adapted  to  logarithmic  cal- 
culations of  the  angles  when  the  sides  are  given. 

(2)  To  derive  tan  |  J,  tan  ^  e,  tan  ^  a.  Values  for  tan  ^  J, 
tan  |-  <?,  tan  -|  a,  may  be  derived  directly  from  (5),  (6),  (7), 
§  86,  by  manipulation  similar  to  that  employed  for  tan  |  B, 
tan  ^  C,  tan  ^  A. 

Another  method  consists  in  transforming  (8),  (9),  and  (10) 
directly  into  the  required  values  of  tan  ^  b,  tan  ^  c,  tan  |^  a  by 
means  of  the  polar  triangle.     In  (8)  let  us  substitute 

^=180° -5',  a  =  180° -A',  J  =  180° -5',  c  =  180°-6", 

1^=90°-!  J',  s  =  l(a+b  +  c}  =  210°->S, 

s- h  =  90° -(S-B'),  s-e=90°-(S-O'}, 

s-a=90°-(S-A'), 
where 

S=l(A'+B'+C')>90. 

Then  we  have 


tan  (90°  -  1  6')  =  xh  ^^^^°  "  ^^-  ^^>^  ^"^  1"^^°  "  ^'^~  ^"^"l 
^  2     ^      \         sin(270°-AS')sin[90°-(*S~j5')] 

or,  

cot  1  ^^_Jcos(^-^^)cos(^-CO 
cot  2  6-^      -cos^cos(^-^')    ' 

with  similar  expressions  for  cot  ^  c',  and  cot  ^  a'. 

Taking  the  reciprocal,  and  omitting  accents,  we  have 

^     ^  2         ^'cos  (S  -  A)  cos  (S  -C)  ^  " 


§§  87-8b]         THE   OBLIQUE   SPHERICAL   TRIANGLE  131 


(12)  imlc  =  J     -  g>^  ^ «o^  ('^ -  ^\=K cos  (S  -  C), 
^     ^  ^         ^cos  (S  —  B)  cos  (S  -  A) 

(13)  tmla  =  V-=7f^  ^^^'  ^^~  ^^»,  =  ^ cos  (^  -  A), 

^     ^  ^         ^cos  (/S— C)  cos  (/S  — £) 


1i---x  -cos,S' 


'cos  (.S  -  A)  cos  (*S  -  B)  cos  (-S  -  C; 

Note.  It  may  be  shown  that  k,  used  in  (8),  (9),  (10),  is  equal  to  the 
tangent  of  the  radius  of  the  small  circle  inscribed  in  the  spherical  triangle 
ABC,  and  that  K,  used  in  (11),  (12),  (13),  is  equal  to  the  tangent  of  the 
radius  of  the  small  circle  circumscribing  the  triangle  ABC. 

88.  Napier's  Analogies.  Dividing  tan  |  B  by  tan  |  C,  for- 
mulas (8),  (9),  §  87,  we  have 

tan  \B _  sin (s  —  c) 
tan  I  O'      sin(s  — 6) 

Taking  this  equality  by  composition  and  division, 

tan  \  B  +  tan  \G  _  sin  (%  —  c)  -\-  sin  (s  —  5) 
tan  I  B  —  tan  |  (7     sin  (s  —  (?)  —  sin  (s  —  5) 

which  reduces  to 

sin  I  (B+  0)  _  sin  I (2s—h-e)  cos  | (5  -  c) 
sin  I  (^  —  O)      cos  1(2  8  —  6  —  c)  sin  g^  (^  —  c) 
and  finally, 

sills  rB  +  C)         tanjrt 

(14)  —i =  — i-= 

^  sills  (B-C)     tan.](&-c) 


Similarly, 
(15) 

(16) 


Hinl(C  +  A)  tanift 


i  ~        1  • 

sin-^(C  —  A)     tan.,  (c-rt) 

8in^(^  +  J5)  tan^c 


sinl(A-B)     tan|(a-6)* 

Another  formula  may  be  obtained  by  multiplying  tan  ^  B  by 

tanA^tani(7='5ill^i^:^. 


tan  I  a 


sin  8 


132  SPHERICAL   TRIGOXOMETRY  [§88 

Write  tan  ^  B  and  tan  }^0  in  terms  of  sines  and  cosines,  and 
take  the  equation  by  division  and  composition, 

cos  ^  B  cos  \  C  —  sin  ^  B  sin  \  O _  sin  8  —  sin  («  —  a) 
cos  ^  ^  cos  \C  -{-  sin  ^  ^  sin  \  C     sin  «  +  sin  (»  —  a) 

and  reduce,  giving 

C08|(B+C)  tanja 


(17) 

Similarly, 

(18) 

(19) 


2^        1    ^J  _         '■■»"2^ 

cosJ(B-C)     taiij(6+c) 


eo%\{C  +  A)         tan*  6 


cosi(C-^)     tan^(c  +  a) 

cos  \  {A  +  B)         tan  \  c 
cos  ^  (^  -  B)     tan  \{a  +  h) 


The  formulas  (14)  to  (19)  are  known  as  Napier's  Analogies. 
Numbers  (14)  and  (17)  enable  us  to  compute  the  sides  6,  c, 
when  the  opposite  angle  B^  (7,  and  the  included  side'  a  are 
known.  The  Theorem  of  Sines,  §  84,  then  determines  the 
remaining  angle  A. 

Napier's  Analogies  appear  also  in  another  form  which  may 
be  derived  by  means  of  the  polar  triangle. 

By  substituting  a  =  180°  -A\A  =  180°  -  a',  and  so  on,  let 
the  student  deduce  from  (14)  to  (19)  the  following : 


(20) 

(21) 

(22) 
and 
(23) 

(24) 


sin  \{b  +  c)  cot  5  A 


sm^(&-c)  tan|(B-C) 

sin  ^{c  +  a)  cot  |  ^ 

8ini(c-a)  tani(C-4)' 
sin  \{a  +  b)  cot  \  C 


8in|(a-6)  tani(^-^)' 

cos  I  (6  +  c)  cot  2  ^ 

cos  lib-c)  tan  I  (B  +  C) ' 

cos  5  (c  +  a)  cot  I  B 


cosi(c-a)     tan^(C+^) 


§§88-90]         THE   OBLIQUE   SPHERICAL   TRIANGLE  133 

cos  I  (a  +  b)  cot  I  C 


(25) 


cos\ia  —  b)     tanliA  +  B) 


89.  The  Area  of  a  Spherical  Triangle.  The  sura  of  the  three 
angles  of  a  spherical  triangle  is  greater  than  180°.  If  A,  B,  C 
be  the  angles,  A  +  B+  C-1S0°  =  U 

determines  the  spherical  excess  of  the  triangle. 

In  spherical  geometry  it  is  shown  that  the  areas  of  spherical 
triangles  are  to  each  other  as  their  spherical  excesses.  A  tri- 
rectangular  triangle  has  an  area  |  7ri2^  and  its  excess  is  90°. 
Hence,  if  A  denote  the  area  of  any  spherical  triangle  whose 
spherical  excess  is  U°,  and  R  be  the  radius  of  the  sphere,  we 
have 

Ai^irB^-.-.U-.dO, 
or 

180 

The  spherical  excess  U  is  readily  found  when  the  angles  of 
a  spherical  triangle  are  known. 

The  value  of  U  may  be  determined  directly  when  the  three 
sides  are  known.     The  resulting  formula  is  known  as  L'Huiliers 
Theorem^  and  is  given  by 
E 


tan  —  =  Vtan  ^8  tan  |^(s  —  a)  tan  |(8  —  i)  tan  |(s  —  c). 

The  derivation  of  this  formula  will  not  be  given  here. 

90.  Solution  of  Oblique  Spherical  Triangles.  When  any  three 
parts  of  a  spherical  triangle  are  given,  the  remaining  parts  may 
be  found  by  use  of  the  Theorem  of  Sines,  the  Half -angle 
Formulas,  or  by  Napier's  Analogies.     Six  cases  occur. 

Case  I.  G-iven  the  three  sides,  a,  J,  c.  To  determine  the 
angles  A,  B,  C,  we  use  the  half-angle  formulas 

tanlj.  = ,  t'dn^B=- —,  t&n^C=- — -, 

^  sin(s-a)  ^         sin(8-J)  ^         sm  («  —  <?)' 


where  k  =  J^'^n  ( s  -  a) sin  (s  -  b) sin  js  -  c) , 

^  sin« 

Employ  logarithms  in  the  calculation,  tabulating  work  after 
the  form  suggested  in  the  solution  of  plane  triangles. 


134  SPHERICAL   TRIGONOMETRY  [§  90 

Case  II.      Given  the  three  angles.  A.,  B,  C.     Determine  the 
sides  a,  6,  c  by  use  of 

tan  ^  a  =  ^cos  (^S  —  A), 

tan  \c  =  Kcos  {S  —  C), 
where  K=^-  ~     ^ 


cos  (S  -  A)  cos  {S  -  B)  cos  (^S  -  0) 
In  this  case  cos  S  is  always  negative. 

Case  III.  Given  two  sides  and  the  included  angle,  c,  a,  B. 
To  find  C  and  A  use  Napier's  Analogies : 

tani(C+A)  =  22iM£^coti£, 
COS  ^  (c  +  a)  , 

tan  K^-  ^)=  ^!°  f^""  ~  ""^  cot  1^. 

In  the  first  of  these  formulas  cot  |  B  is  positive,  cos  |(<7  —  a) 
is  positive ;  if  cos  |^  (e  +  a)  be  positive,  the  tan  ^  (  (7+  A}  is 
positive,  and  |^  (  0+  A}  <  90°.  If  cos  ^  (c  +  a)  be  negative,  the 
tan  i((7+ J.)  is  negative,  and  1(C+ J.)>  90°. 

After  the  angles  C  and  A  have  been  found,  the  side  b  may 
be  determined  by  means  of  the  Theorem  of  Sines, 

sin  a  sin  B 


sin  h  = 


sin^ 


In  using  the  Theorem  of  Sines  care  must  be  exercised  in 
determining  whether  b  lies  in  the  first  or  second  quadrant. 

Case  IV.  Given  two  angles  and  the  included  side,  B,  C,  a. 
Formulas  (17)  and  (14)  of  Napier's  Analogies  determine  b 
and  c.  irn     r<\ 

The  first  of  these  formulas  shows  that  if  \(^B  +  C')>  90°,  then 
will  I  (J  +  c)  >  90°,  and  inversely.  The  angle  A  may  be 
determined  by  the  Theorem  of  Sines. 


§90]  THE   OBLIQUE   SPHERICAL   TRIANGLE  135 

Case  V.      Given  tivo  sides  and  an  angle  opposite  one  of  them, 
b,  (?,  B.     Here,  the  angle  C  is  given  by 
sin  c  sin  B 


sin 


C  = 


sin  b 

(1)  If  sin  c  sin  B 
>  sin  b,  no  solution 
exists. 

(2)  If  sin  c  sin  B 
=  sin  b,  the  angle 
C  =  90°. 

(3)  If  sine  sin  ^ 

<  sin  b,  either  one  or  two  solutions  will  occur;  see  Fig.   60. 
After  C  has  been  found,  angle  A  and  side  a  may  be  obtained 

Case  VI.  Given  two  ayigles  and  a  side  opposite  one  of  them, 
B,  C,  b.     The  side  c  is  given  by 

sin  b  sin  O 

sin  c  = : — -—  • 

sin  B 

This  case  may  also  have  no  solution,  one  solution,  or  two 
solutions.  After  e  has  been  determined  the  remaining  parts, 
a.  A,  may  be  derived  from  Napier's  Analogies,  as  in  Case  V. 

EXERCISES 
Solve  the  following  spherical  triangles  for  the  unknown  parts: 

1.  ^  =  100°,   B  =  15°,    0=65°. 

2.  a  =  27°  40',  6=48°,    c=50°40'. 

3.  a  =  65°  48',  J  =  120°  21',    c=84°21'. 

4.  ^=96°  50',  ^=75°  10',    (7=  96°  50'. 

5.  5  =  72°  30',  ^  =  41°  27',   (?  =  49°17'. 

6.  6  =  118°  48',   (?=71°24',   A  =51°  24'. 

7.  a  =72°  48' 12",   6  =  41°  38' 10",    (7=  33°  24' 10". 


136  SPHERICAL   TRIGONOMETRY  [§90 

8.  ^  =  129°  24' 10",    (7=  31°  24' 30",   a  =  42°  50' 40". 

9.  a  =  112°  40' 15",   J  =  121°  10' 10",    C=  128°  44' 15". 
10.  ^  =  100°  14' 20",   J?  =  140°  40' 40",   c=  39°  28' 15". 
n.    5  =  54° 21',   c?=31°48',   ^=100°  10'. 

12.  ^  =  101°  40',   (?  =  62°21',   a  =104°  24'. 

13.  5  =  40°  16' 10",   c?  =  46°  48' 40",   ^  =  56°  21' 50^'. 

14.  a  =  37°  10' 20",   6  =  112°  48',   ^  =  34°  28' 15". 

15.  (7=  123°  38',   6  =  150°  40',   c  =  10°  21' 30". 

16.  ^  =  140°  29',    6^=  24°  40',   a  =  135°  40'. 

17.  5  =  128°  40',    (7=  141°  25',   c=125°47'. 

18.  ^=48°37',    (7=34°48',   e  =  41°10'. 

19.  ^=99°  27',    C  =  89°21',   a  =  45°  37'. 

20.  a  =  71°  47',   6  =  145°  47',    (7=  130°  50'. 

21.  When  h  =  37°  28',  c  =  65°  21',  and  A  =  87°  40',  find  side  a 
directly  by  Modified  Formula  {B),  §  85. 

22.  Find  c  directly,  (B)  §  85,  when  a  =  101°  48',  J  =  100°, 
(7=  68°  41'. 

23.  Find  in  inches  the  perimeter  of  a  spherical  triangle  with 
sides  68°,  96°,  120°,  on  a  sphere  whose  radius  is  15  in. 

24.  Find  the  perimeter  of  a  spherical  triangle  with  angles 
69°,  84°,  100°,  upon  a  sphere  whose  radius  is  10  in. 

25.  Find  the  area  of  the  spherical  triangle  with  angles  120°, 
84°,  72°,  upon  a  sphere  whose  radius  is  20  in. 

Find  the  areas  of  the  following  spherical  triangles: 

26.  A  =24°  30',    ^=140°,    C=80°,   radius  =12  in. 

27.  a  =  65°,    5  =  47°,    e=60°,   r  =  15in. 

28.  ^  =  100°,   ^=80°,    c=60°,   r  =  12in. 

29.  ^.  =  37°,   c=84°,   A  =  65°,   r=  3956  mi. 

30.  5  =  104°,    6^=128°,   6  =  138°,   r=  3956  mi. 


§  91]  LATITUDE   AND   LONGITUDE  137 

APPLICATIONS  OF   SPHERICAL   TRIGONOMETRY 

91.    The   Earth  as  a   Sphere.     Definitions  and  Notation.     In 

what  follows  the  earth  is  assumed  to  be  a  sphere  with  a  radius 
of  3956  mi.  The  shortest  distance  between  two  points  upon 
the  earth  is  the  length  of  the  shorter  arc  of  a  great  circle 
drawn  through  the  two  points. 

(1)  The  Creographical  Mile  and  Statute  Mile.  The  geo- 
graphical mile  (called  nautical  mile,  also  knot}  is  a  unit  of  dis- 
tance along  the  arc  of  a  great  circle  upon  the  earth ;  it  is  the 
length  of  an  arc  of  one  minute  of  a  great  circle,  hence  sixty 
geographical  miles  equal  one  degree.  The  statute  mile  is  our 
ordinary  unit  of  measurement  equivalent  to  6280  ft.  The 
length  of  one  degree  of  the  arc  of  a  great  circle  of  the  earth 
is  given  by 

-,o         TT  X  3956        .  o   i/<1  CO 

1    = mi.,  TT  =  3.14159. 

180 

(2)  Meridian,  Longitude,  Latitude.  A  great  circle  drawn 
through  the  poles  iV,  S,  and  through  any  point  B,  Fig.  61,  is 
called  the  Meridian  of  the  point  B. 

Meridians  are  numbered  to  the  east  and  west  with  reference 
to  some  zero  meridian.  The  meridian  passing  through  Green- 
wich (near  London),  England,  is  usually  taken  as  the  zero 
meridian,  NGiPS,  Fig.  61.  The  meridian  of  Greenwich  inter- 
sects the  earth's  equator  LJLMW  in  P. 

The  Longitude  of  any  point  B  upon  the  earth  is  the  number 
of  degrees  between  the  zero  meridian  and  the  meridian  through 
B.  This  angle  is  measured  by  the  arc  of  the  equator  PEM, 
or  the  spherical  angle  GrNB.  The  longitude  of  F  is  the  arc 
PEL,  or  the  angle  GNF. 

The  meridians  are  numbered  by  degrees,  minutes,  and  sec- 
onds from  0°  to  180°  east  (marked  E.)  and  west  (marked  W.) 
of  the  Cireenwich  or  zero  meridian.  Thus  the  point  B  is  in  W. 
longitude,  also  the  point  F. 

The  Latitude  of  a  point  B  is  the  arc  of  the  meridian  of  the 
point  intercepted  by  the  equator  and  the  point.  The  latitude 
of  B  is  the  arc  MB.  Latitude  is  counted  positive  or  negative 
according  as  the  point  is  north  or  south  of  the  equator ;  the 


138 


SPHERICAL  TRIGONOMETRY 


[§  91-92 


direction   is  usually  indicated   by  attaching  N.  or   S.  to   the 
number  of  degrees  in  the  latitude. 

As  samples  of  the  notation  of  latitude  and  longitude  we  may 
write : 


(1)  New  York, 

(2)  Boston, 

(3)  Greenwich, 

(4)  Liverpool, 


lat.  40°  43'  N.,  long.  74°  W. 
lat.  42°  21'  N.,  long.  71°  4'  W. 
lat.  51°  29'  N.,  long.  0°. 
lat.  53°  24'  N.,  long.  3°  4'  W. 

(5)  San  Francisco,  lat.  37°  48'  N.,  long.  122°  28'  W. 

(6)  Valparaiso,        lat.  33°  2'    S.,  long.  71°  41'  W. 

(7)  Calcutta,  lat.  22°  33'  N.,  long.  88°  19'  E. 

(8)  Sandy  Hook,     lat.  40°  28'  N.,  long.  74°  V  W. 

92.    The  Terrestrial  Triangle.     When  the  latitude  and  lonf/i- 
tude  of  any  two  points  upon  the  earth  are  known,  the  distance 


between  the  points  may  be  found.  Thus,  in  Fig.  61,  if  the 
point  B  be  in  lat.  «°  N.,  long.  y8°  W.,  and  F  be  in  lat.  ai°  N., 
long.  /8j°  W.,  the  spherical  triangle  NFB  is  completely  deter- 
mined as  follows ; 


§  92]  LATITUDE   AND   LONGITUDE  139 

arcNB  =  90^  -  a^  =  complement  of  the  latitude  of  B, 
arc  JV"F=  90°  —  ai°  =  complement  of  the  latitude  of  1^, 
angle  BNF  =  Pi°  -  P^  =  difference  in  longitude  of  B  and  F. 

This  data  gives  two  sides  and  the  included  angle  of  the  triangle 
NFB. 

(1)  To  find  the  distance  between  two  points  whose  latitude  and 
longitude  are  given.  If  the  distance  BF,  Fig.  61,  alone  be  re- 
quired, we  have  to  solve  the  spherical  triangle  BNF  for  BF. 
This  may  be  done  by  the  use  of  (-B),  §  85,  in  which  logarithmic 
calculation  may  be  employed. 

J) XI     GosNF       .    ./J  ,    TiT-DN      since,  ^        a^ 

cos  BF  =  — : — --  X  sin(^  +  JVB)  = J  x  cos(a  —  ^), 

sin  6  sin  a  ^ 

.       a        cot  NF             tan  a, 
tan  C7  = = 1 — ■  • 

cos  BNF      cos(;8i-/S) 

The  arc  BF  may  also  be  obtained  by  use  of  the  Cosine 
Theorem  (2),  §  85, 

cos  BF  =  cos  NB  cos  JVF  +smN~B  sin  NF  cos  BNF, 
=  sin  a  sin  «j^  +  cos  a  cos  a^  cos  (ySj  —  /8), 

but  the  use  of  this  formula  is  not  to  be  recommended  except 
perhaps  when  the  angles  are  such  that  the  products  sin  a  sin  a^ 
cos  a  cos  «j  cos(ySj  —  /3)  reduce  readily  by  ordinary  multipli- 
cation. 

(2)  To  find  the  bearing  and  distance  of  two  points  whose  lati- 
tude and  longitude  are  known.  The  angle  NBF,  Fig.  61,  is  the 
bearing  of  F  from  B  ;  the  angle  NFB  is  the  bearing  of  B  from 
F.  The  bearing  of  a  course  at  a  given  period  is  usually  con- 
sidered as  the  smaller  angle  which  that  course  makes  with  the 
meridian  through  the  point.  Thus  in  the  figure,  the  bearing 
of  F  from  B  is  N.  7°  W.,  if  7  =  Z  NBF<  90° ;  the  bearing  of  F 
from  B  is  S.  7^°  W.,  if  7  >  90°  and  7^  =  Z  SBF=  180°  -  7. 

When  the  bearing,  or  bearing  and  distance,  of  two  points 
whose  latitude  and  longitude  are  given,  is  required,  Napier's 
Analogies,  (20),  (23),  §  88,  may  be  used  to  find  the  angles 
NBF  and  NFB.  The  side  BF  may  then  be  found  by  use  of 
the  Theorem  of  Sines,  (1),  §  84. 


140  SPHERICAL   TRIGONOMETRY  [§§  92-93 

EXERCISES 

1.  Find  the  distance  in  knots  between  Boston,  lat.  42°  21'  N., 
long.  71°  4'  VV.,  and  Liverpool,  lat.  53°  24'  N.,  long.  3°  4'  W. 
Find  also  the  bearing  of  each  port  from  the  other. 

2.  Find  the  distance  in  statute  miles  along  the  arc  of  a 
great  circle  from  New  York,  lat.  40°  43'  N.,  long.  74°  W.,  to 
San  Francisco,  lat.  37°  48'  N.,  long.  122°  28'  W. 

3.  Find  the  distance  in  nautical  miles  from  New  York, 
lat.  40°  43'  N.,  long.  74°  W.,  to  Greenwich,  lat.  51°  29'  N., 
and  the  bearing  of  each  point  from  the  other. 

4.  If  a  vessel  sails  directly  east  from  Sandy  Hook,  lat. 
40°  28'  N.,  long.  74°  1'  W.,  along  the  arc  of  a  great  circle  at 
the  uniform  rate  of  16  knots  per  hour,  find  its  latitude  and 
longitude  (1)  at  the  end  of  48  hours'  sailing,  (2)  at  the  end  of 
five  days'  sailing. 

5.  Find  the  distance  in  statute  miles  from  San  Francisco, 
lat.  37°  48'  N.,  long.  122°  28'  W.,  to  Calcutta,  lat.  22°  33'  N., 
long.  88°  19'  E.,  and  the  bearing  of  each  point  from  the  other. 

93.  The  Celestial  Sphere.  Astronomical  problems  furnish 
many  applications  of  spherical  trigonometry.  One  class  of 
these  problems  will  be  noticed  here. 

The  daily  rotation  of  the  earth  upon  its  axis  from  west  to  east 
Causes  the  stars  to  seem  to  rotate  from  east  to  west  upon  the 
surface  of  an  immense  sphere  named  the  celestial  sphere.  To  a 
person  located  at  any  point  upon  the  earth  one-half  of  this 
sphere  is  visible.  The  celestial  sphere  is  represented  in  Fig.  62, 
the  earth  being  a  mere  point  at  the  centre. 

(1)  The  Horizon  of  any  point  upon  the  earth  is  the  intersec- 
tion of  the  horizontal  plane  through  the  point  with  the  celes- 
tial sphere.     In  Fig.  62,  the  horizon  is  the  great  circle  HLH' . 

(2)  The  Zenith  of  any  point  is  the  intersection  of  the  perpen- 
dicular erected  to  the  plane  of  the  horizon  at  the  point.  The 
point  on  the  celestial  sphere  diametrically  opposite  the  zenith  is 
the  Nadir.     In  the  figure  Z  is  the  zenith,  Z'  is  the  nadir. 

(3)  The  Celestial  Poles  are  the  intersections  of  the  line  of 
the  earth's  axis  with  the  celestial  sphere.     The  point  iV  is  the 


§§  93-94] 


THE   CELESTIAL   SPHERE 


141 


north  pole  of  the  celestial  sphere,  *S'  is  its  south  pole.  The 
celestial  sphere  rotates  (apparently)  about  the  axis  NS  once  in 
24  hours. 

(4)  The  Celestial  Equator  is  the  intersection  of  the  plane  of 
the  earth's  equator  with  the  celestial  sphere.  In  the  figure, 
EME'  is  the  celestial  equator,  N  and  S  are  its  poles. 


(5)  The  Celestial  Meridians  are  the  great  circles  through  N 
and  S.  The  celestial  meridian  through  any  star  is  called  the 
hour  circle  of  that  star.     Thus,  NPMh  the  hour  circle  of  P. 

94.  The  Celestial  Triangle.  The  position  P  of  any  star  is 
known  if  we  know  its  distance  PL  above  the  horizon,  and  its 
distance  PiUf  north  or  south  of  the  celestial  equator. 

(1)  The  Altitude  PL  of  a  star  at  P  is  the  arc  of  a  zenith 
circle  ZL  intercepted  between  the  point  P  and  the  horizon. 
See  Fig.  62. 

(2)  TJie  Declination  of  a  star  P  is  the  arc  PM  oi  the  celes- 
tial meridian  iVPillf  between  the  point  and  the  celestial  equator. 

(3)  The  Latitude  of  the  Zenith  of  any  point  on  the  earth  is 
the  arc  EZ  and  is  equal  to  the  latitude  of  the  observer. 

(4)  The  Hour  Angle  of  any  star  is  the  angle  between  the 


142  SPHERICAL   TRIGONOMETRY  [§94 

meridian  through  the  star  and  the  zenith   meridian.     In   the 
figure  PNZ  is  the  hour  angle. 

Since  the  celestial  sphere  apparently  rotates  through  360°  in 
24  hr.,  or  15°  in  one  hour  of  time,  we  may  express  the  hour 
angle  in  hours,  minutes,  and  seconds,  and  thus  determine  the 
time  required  for  a  star  to  rotate  into  the  meridian  of  the 
zenith. 

The  points  P,  iV,  Z  determine  the  celestial  triangle  PNZ, 
sometimes  called  the  astronomical  triangle.  This  triangle  is 
determined  when  the  latitude  of  the  observer  EZ,  the  altitude  of 
the  star  PL,  and  the  declination  of  the  star  PM  are  known. 
We  have : 

NZ=  90°—  latitude  of  the  observer, 

ZP  =  90°  -  altitude  of  the  star, 
PN=  90°—  declination  of  the  star. 

If  an  observation  of  the  sun's  altitude  be  made  in  the  fore- 
noon, let  us  sky,  and  the  declination  of  the  sun  upon  that  day 
of  the  year  be  known,  then  we  may  compute  the  hour  angle 
provided  the  latitude  of  the  observer  be  known.  Having  the 
hour  angle,  we  may  find  the  corresponding  equivalent  in  time, 
and  thus  determine  the  hour  of  day  at  which  the  observation 
was  made. 

The  sun's  declination  varies  from  near  23°  30'  south  to  near 
23°  30'  north  declination.  At  the  vernal  equinox  and  at  the 
autumnal  equinox  the  sun's  declination  is  zero. 

EXERCISES 

1.  At  San  Francisco,  lat.  37°  48'  N.,  a  forenoon  observation 
shows  the  sun's  altitude  as  40°  21'.  If  the  sun's  declination  be 
10°  41'  N.,  what  is  the  time  of  observation  ? 

2.  In  latitude  40°  21'  N.,  the  sun's  altitude  in  the  afternoon 
was  found  to  be  35°  40'.  What  was  the  time  of  observation, 
if  the  sun's  declination  is  8°  4'  S.? 

3.  Find  the  time  of  sunrise  at  Boston,  lat.  42°  21'  N.,  the 
sun's  declination  being  20°  31'  N. 

4.  Find  the  time  of  sunrise  (approximately)  at  a  point  whose 
latitude  is  39°  10'  N.  twenty  days  after  tbe  vernal  equinox. 


CHAPTER   XII 

FORMULAS 

PLANE   TRIGONOMETRY 

Fundamental  Identities. 

1.  sin  X  CSC  X  =  cos  x  sec  x  =  tan  x  cot  a;  =  1. 

2.  sin^ a;  +  cos^ a;  =  1,    sec^ a:  —  tan^ a;  =  1,    csc^a;— cot2a;  =  1. 

sin  X        1         sec  x      Vl  —  cos^  x 

3.  tana;  =  ^ = — — -  = = 

cos  a;      cotar      esc  a;  cos  a; 

1  tana; 


sin  X  =  cos  X  tan  x  = 
cos  X  =  sin  X  cot  x  = 


Vl  4-  cot^  X      Vl  +  tan^  x 
1  cot  a: 


Vl  +  tan^a;      Vl  +  cot^x 

Sum  and  Difference  Formulas. 

6.  sin  (a;  ±  ?/)  =  sin  a;  COS  z/ ±  COS  a;  sin  y. 

7.  cos(a:  ±y)  =  COS  a;  COS  «/ T  sina;  sin  y. 

.       .         V        tan  X  ±  tan  y 

8.  tan  Cx  ±  y)  = ^  • 

It  tan  a:  tan  ^ 

cot  X  cot  y  T  1 

9.  cot(a;±?/)  =  — ; — ^-: 

^       ^  cot  y  ±  cot  X 

10.  sin  a;  +  sin  y  =  2  sin  1^ (a:  +  3/)  cos  |^(a;  —  y). 

11.  sin  a:  —  sin  ?/  =  2  cos  |(a: 4-  «/)  sin  -|  (a:  —  y). 

12.  cosa;  +  cos2/=  2cos|^(a;  +  y)cos|(a;— ?/). 

13.  cos  a;  —  cos  y  =  —  2  sin  -i-  (x  +  y)  sin  |  (x  —  y). 

sin  (x±y^ 

14.  tan  a;  ±  tan  y  =  — - — — -— -  ■ 

^      cos  X  cos  ^ 

sin  (x  ±  y) 

15.  cota;±coty  =  ±— T ■. 

"^  sin  a:  sin  y 

sin  a:  4-  sin  w      ,       .  ,     ,     . 

16.    ■ ^  =  tani(a;  +  y). 

cos  X  +  COS  y 

143 


144  FORMULAS 

sin  X  4- sin y  _  tan  \{x  +  'if) 
sin  a;— sin  ^      Vdca.\(x  —  y) 

18.  sin^  X  —  sin'^  y  =  sin  {x  +  2/)  sin  (a:  —  y^. 

19.  cos^  a;  —  cos^  y  =  —  sin  (^x  +  y')  sin  (x  —  y^. 

20.  cos^  a;  —  sin^  ?/  =  cos  (a;  +  ^)  cos  (x  —  y^. 

Half  Angle  and  Multiple  Angle  Formulas. 

2  tan  i  X 


21.    sin  a;  =  2  sin  1^  a;  cos  \x  = 


1  +  tan2| 
2 


22.  COS x  =  cos^  1 2^  —  sin^ ^- a;  =  2  cos^ |^a;  —  1=1  —  2 sin^  ^ 

1  —  tan^  1  X 
1  +  tan^  ^  X 

2  tan  i  a;  2  cot  i  a;  2 

23.  tan  X  =  3 i-i —  = 


1  —  tan^  1 X      cot^  1^ a;  —  1      cot ^x  —  tan  |  a; 
24.    cota;  =  ^-^Jp^  =  l(cotia;-tanla:). 


25.  sin|  a:  =  V^l  —  cosa;). 

26.  cos^  a;=  Vi(l  +  cosaj). 


2  •^—  "  2 
27.    tan|^a:  =  v- 


—  cos  X         sin  a:  i  —  cos  x 


+  cosa;      1  +  cosa;  sin  a; 

28.  sin  2  a;  =  2  sin  a;  cos  x,  cos  2  a;  =  cos^  x  —  sin^  a;. 

29.  sin  3  a;  =  3  sin  a;  —  4  sin^  x,  cos  3  a;  =  4  cos^  a;  —  3  cos  x. 

30.  sin  4  a;  =  sin  a;  (8  cos^  a:  —  4  cos  a;), 
cos  4  a;  =  8  cos*  a;  —  8  cos^  a;  +  1. 

„,      ,       o  2  tan  a;         .       «         3  tan  a:  —  tan^  a; 

31.  tan  2 a;  = r— ,    tan  6x  =  — 

1  —  tan^  X  1  —  3  tan^  x 

Plane  Triangles. 

„^     rr^^  £  •  sin  A     siu  B     sin  C        1 

32.  i  neorems  oi  sines  : =  — ; —  = =  — —  • 

a  h  c         2R 

33.  Theorem  of  cosines  :   aP"  =  h^  -{-  <P-  —  1  he  cos  A. 

34.  Theorem  of  tangents :  •  ' 

a  +  l  __  sin  A  +  sin  B  _  tan  ^  ( J.  +  ^) 
a  —  h      sin  A  —  sin  B      tan  ^  (J.  —  _S) 


FORMULAS 


145 


35.    The  half  angles 
(1>  sin 


^"^=V^ be 


(2)  cos-=V-V^' 

(3)  tan4  =  JSESCZEZ) 
^  ^  2       ^      «(s-a) 


►,  where  2«  =  a  +  5  +  (?. 


sin  A  sin  jB 


36.    Area  :  K  =lhc  sin  A  =  A^  c^  ,       _ 

^  ^      sin  (J.  +  5) 


=  Vs  (s  —  «)(s  —  J)(s  —  c)  =  «r. 


Analytic  Trigonometry. 
37.    Inverse  identities : 


(1)    sin"^  a;  ±  sin~i  z/ =  sin  ^  (x^l  —  y^  ±yy/l—3p''). 


(2)  cos"^2- ±  cos'^  ?/ =  cos  1  (a;y  T  Vl  —  a^Vl  —  ^2). 

(3)  tan-i  ^  ±  tan-i  y  =  tan"!  f  f  ^  ^  ^ . 

\lTxyJ 

(4)  sin"i  a^  =  ^  sin"i  (2  x  VI  —  a:^)  =  i  tt  —  cos"^  a; 
1 


=  CSC 


-1     _ 


—  sin  1  (  —  a;). 


(5)    cos"i  a;  =  ^  cos"i  (2  a;2  —  1)  =  i  TT  —  sin"i  a; 


=  2  tan-i 


^1+x 


.-il 


sec"^-. 

X 


38.    sin  a;  =  — -,  («''' 
2  I 

1/e^-f?- 
tan  a;  =  - 


e"'^),  cos  a;  =  -  (g"^  +  e  "), 


39.  sinha:  =  |^(e-^  — e"^),  cosh  a;  =  |(e^  +  ^"0' 

tanh  a:  = ^ . 

e^  +  e  ^ 

40.  e'-^  =  cos  x-\-  i  sin  a;,  g "'^  =  cos  x  —  i  sin  x. 

41.  e^  =  cosh  X  -\-  sinh  a;,  e"*-^  =  cosh  x  —  sinh  x. 

42.  sin(za;)  =  i(g^  — e'O  =  *s^'^^^' 


cos  (ia;)  =  1  (g-^  +  e  -•')  =  cosh  a;,  tan  (za;)  =  i 


.e-^  —  e 


e^  +  e' 


—  %  tanh  x. 


146 


FORMULAS 


43.  (cos  x-\-  i  sin  a:)"  =  (e'-^)"  =  e"'^  =  cos  nx  +  i  sin  nx. 

44.  sin  (x  ±  iy)  =  sin  x  cos  (i^}  ±  cos  x  sin  (ly)  ^ 

=  sin  a;  cosh  y  ±i  cos  a;  sinh  y, 

45.  cos(a;  ±  iy)a;  =  cos  x  cosh  y  T  *  sin  x  sinh  y. 

46.  cosh^z  —  sinh^  =  sech^a;  +  tanh%  =  coth^a;  —  csch^a;  =  1. 

47.  Infinite  series : 

(1)    (l  +  a:)»==l  +  na:  +  ^^^^>a^+<^-^X^-^)a:3^.., 

/y*^  /yO  /**4  /y»U 

^v^  /y^  /yrt  rt/^ 

(3)  log(l  +  a:)  =  a:-f  +  |-f +  ?-.... 

2        d        4        O 

/>«o  /*^  /*»( 

(4)  sm.  =  .-|-^+---+.... 

a^      />j4      /»^ 

(5)  COS.=  l--+g-|^+.... 
(6)   tan.  =  ^  +  -  +  3^  +  ^j^+.... 

^«3  rjM>  rjri 

(7)  sinha;  =  a:  +  -^  +  |^+iY+  -. 

[3     [5      |7 

/ywa  /vot  /y«D 

(8)  cosh.  =  1+1  +  1  +  1+.... 

3^      2  a^ 

(9)  tanha;  =  a;— —  +  — •••. 

o         J.0 


48.    sin  X 


49.    cos  a; 


=^{-(lT}{-feT}{-fe)T- 
=l-(^71{-(l-:T){-(.lf)T- 


50.    If  sec  ^  =  cosh  X,  0  =  gudermannian  of  a:  =  ydx. 

(1)  6  =  gdx  =  sec~i  (cosh  a;). 

(2)  X  =  gd-^  =  cosh-i  (sec  6). 


FORMULAS  147 

SPHERICAL   TRIGONOMETRr 
The  Right  Spherical  Triangle. 

51.    sinvl  =  ?i^.       52.    cos^  =  *^.      53.    tan  ^  =  *5ILf . 
sm  c  tan  c  sin  h 

54.  cos  0  =  cos  a  cos  h  =  cot  A  cot  ^. 

55.  cos  A  =  sin  ^  cos  a. 

56.  Napier's  Rules ; 

(1)  sin  (mid.  pt.)  =  tan  (adj.  pt.)  •  tan  (adj.  pt.). 

(2)  sin  (mid.  pt.)  =  cos  (op.  pt.)  •  cos  (op.  pt.). 

The  Oblique  Spherical  Triangle. 
^„     sin  A      sin  £      sin  O 

57.       — ; =   —, -  =  . 

sin  a       sin  b       sin  e 

58.  cos  a  =  cos  b  cos  c  +  sin  b  sin  c  cos  A. 

59.  cos  A  =  —  cos  -B  cos  C  +  sin  jB  sin  (7  cos  a. 

60.  tan^  =  VgilL(i.-^)«"U^-0^ 

2        '      sin  s  sin  («  —  a) 

t      ^  —  ^1    ~  cos /S  cos  (^y  —  J.) 
^"  2  ~  ^^cos  (/S'  -  5)  cos  (S-Cy 

sin  ^  (^  4-  ^)  _        tan  j[g 
sin  I  (J.  —  B)      tan  |^  («  —  ^) ' 
cos  I  (A  +  B}  _       tan  ^  e 
cos  I  (J.  —  ^)  "  tan  1  ( rt  +  J)  * 

g2     sin  I  (a  +  b^  _        cot  ^  O 

sin  1-  (a  —  5)      tan  |^  ( JL  —  5) ' 
cosJ_Ca -f-_6)  _        cot  j  C 
cos  2^  («  —  J)      tan  ^(A  +  B)' 

132 

63.  Area,  A  =  — —  U,  where  B  is  the  spherical  excess. 

lot) 

XT 

64.  tan  —  =  Vtanig  tani(s  -  a)  tani(«  -  6)  tani(«  -  <?). 


61. 


LOGAEITHMIC,    TRIGONOMETRIC, 
AND    OTHER   TABLES 


THE  MACMILLAN  COMPANY 

NEW  YORK   •    BOSTON   •    CHICAGO 
ATLANTA   •    SAN    FRANCISCO 

MACMILLAN  &  CO.,  Limited 

LONDON  •  BOMBAY  •  CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  Ltd. 

TORONTO 


LOGARITHMIC,  TRIGONOMETRIC 


AND   OTHER 


TABLES 


COMPILED   BY 

DAVID   A.   ROTHROCK 

INDIANA    UNIVERSITY 


THE   MACMILLAN    COMPANY 
1911 

All  rights  reserved 


Copyright,  1910, 
By  the  MACMILLAN  COMPANY. 


Set  up  and  electrotyped.     Published  September,  1910.     Reprinted 
January,  1911. 


NorioaoH  ]^(ss 

J.  8.  Cushlng  Co.  —  Berwick  <fc  Smith  Co. 

Norwood,  Mass.,  U.S.A. 


PREFACE 

The  following  Tables  have  been  prepared  to  accompany  the 
author's  "Plane  and  Spherical  Trigonometry."  The  beginner 
in  trigonometric  calculation  should  be  taught  the  use  of  tables 
supplied  with  the  most  convenient  means  of  interpolation.  To 
secure  this  end,  proportional  parts  have  been  inserted  in  the 
Tables  of  Logarithms  of"  Numbers  and  Logarithms  of  the  Trigo- 
nometric Functions.  A  proper  use  of  these  devices  is  especially 
recommended. 

Numerical  calculation  is  a  laborious  process  at  best.  By  the 
professional  computer,  as  well  as  by  the  beginner  in  elementary 
trigonometric  calculation,  every  facility  for  obtaining  the  required 
degree  of  accuracy  with  a  minimum  of  time  and  labor  should 
be  sought.  To  this  end  it  is  recommended  that  (1)  tables  with 
a  minimum  of  decimals  consistent  with  accuracy  be  used,  (2)  for- 
mulas be  arranged  in  a  systematic  manner,  (3)  data  as  well  as 
logarithms  be  systematically  tabulated,  (4)  a  skilful  use  of  pro- 
portional parts  be  acquired,  (5)  logarithmic  calculations  upon 
scraps  of  paper  be  at  no  time  permitted. 


CONTENTS 


Introduction      ..... 

1.  Definitions         .... 

2.  Laws  for  the  Use  of  Logarithms 

3.  The  Co-Logarithm     . 
Explanation  of  Tables 

Logarithms  of  Xumbers,  Table  I 

1.  Common  Logarithms  of  Numbers  1  to  100 

2.  Common  Logarithms  of  Xumbers  100  to  10,009 

3.  Proportional  Parts 

Logarithms  of  the  Trigonometric  Functions,  Table  II 

1.  For  every  second  0"  to  3',  and  89°  57'  to  90° 

2.  For  every  ten  seconds  0°  to  2°,  and  88°  to  90° 

3.  For  every  minute  0°  to  90°,  and  for  0°  to  360'' 

4.  Proportional  Parts  and  Differences  . 

Natural  Trigonometric  Functions,  Table  III 

Miscellaneous  Tables 

1.  Radian  .Measure,  0°  to  180°       . 

2.  Natural  (Napierian)  Logarithms 

(A)  For  integers,  0  to  200 

(B)  For  each  tenth,  1  to  9.9 

3.  The  Hyperbolic  Functions,  sinh  x,  cosh  x 


PAGB 

ix-xiv 
ix 
.  ix-x 
.  x-xi 
xi-xiv 

.  1-20 

1 

.  2-20 

.  2-20 

21-74 
.  22 
23-28 
29-74 
29-74 

75-94 

95-99 
.   96 

97-98 
.  97 
.  98 
.   99 


INTRODUCTION 

Logarithms  may  be  used  to  diminish  the  labor  of  numerical 
calculations  involving  multiplication,  division,  involution,  and  evo- 
lutioyi  by  replacing  these  operations,  respectively,  by  addition, 
subtraction,  multiplication  and  division. 

1.  Definitions.  (1)  The  exponent  by  which  a  number  a  must 
be  affected  to  produce  a  number  N  is  called  the  logarithm  of  X  to 
base  a. 

In  symbols,  if  a*  =  JV, 

log«  X  =  X. 

(2)  There  are  two  systems  of  logarithms  in  use : 

(a)  The  Common  System  (^Briggian  System^  in  which  10  is 
the  base, 

(5)  The  Natural  System  (^Napierian  System^  in  which 
e  =  2.71828-"  is  the  base. 

Examples.        102  =  100,  logiol00  =  2. 

102.23300  ^  171^  logio  171  =  2.23300. 
g3.4oi2o^30^        log,  30  =  3.40120. 

All  ordinary  calculations  are  performed  by  use  of  logarithms 
to  the  base  10,     See  Tables  I,  II. 

Theoretical  calculations  in  the  higher  mathematics  sometimes 
require  the  use  of  natural  logarithms.    See  Tables  IV,  2,  (^),  {B). 

(3)  A  logarithm  of  a  number  usually  consists  of  an  integer 
and  a  decimal.  The  integral  part  of  a  logarithm  is  called  the 
characteristic,  the  decimal  part  is  called  the  mantissa. 

2.  Laws  for  the  Use  of  Logarithms. 

(1)  Law  of  Factors.  The  logarithm  of  a  product  equals 
the  sum  of  the  logarithms  of  the  factors. 

log  (^  X  J5  X  C)  =  log  ^  +  log  B  +  log  C 

(2)  Law  of  Quotients.  The  logarithm  of  a  quotient  equals 
the  logarithm  of  the  numerator  diminished  by  that  of  the  denomi- 
nator. 

log  (^)  =  log.l-logB. 


X  INTRODUCTION 

(3)  Law  of  Involution  (or  Evolution).  The  logarithm 
of  a  7iumber  affected  hy  an  exponent  equals  the  exponent  multiplied 
by  the  logarithm  of  the  number. 

log  {A)^  —  nx  log  A, 

V 

log(^l)'/=^  X  log^. 
Q 

The  laws  of  products^  quotients,  and  involution  (or  evolution) 
hold  for  logarithms  to  any  base. 

(4)  Law  of  the  Characteristic.  The  common  logarithm 
of  a  number  has  a  characteristic  one  unit  smaller  than  the  number 
of  digits  to  the  left  of  the  decimal  point. 

Thus,     log  476.8  =  2.67834,      log  0.4768  =  1.67834, 
log  47.68  =  1.67834,     log  0.04768  =  2.67834. 

In  case  of  a  pure  decimal  the  same  law  of  characteristic  holds, 
the  number  of  digits  being  counted  as  negative.  A  negative 
characteristic  is  usually  indicated  by  a  dash  over  the  integer. 

log  0.04768  =  2.67834. 

In  calculations  involving  negative  characteristics^  10  is  usually 
added,  making  the  characteristic  positive. 

Thus,  log  0.917  =  1.96237  =  9.96237  -  10. 

In  case  of  root  extraction,  it  is  convenient  to  add  10  multi- 
plied by  the  index  of  the  root  to  be  extracted.     Thus, 

log  (0.0732)^=1  xlog  0.0732=^(2.86451)  =  ! (28.86451-30) 
=  9.62150-10  =  1.62150. 

In  this  example,  3  or  any  multiple  of  3  could  have  been  added 
and  subtracted. 

(5)  Law  of  the  Mantissa.  The  mantissce  of  the  loga- 
rithms of  all  numbers  composed  of  the  same  sequence  of  digits  are 
the  same. 

Thus,      log  5297=  3.72403,       log  0.5297  =  1.72403, 
log  52.97  =  1.72403,     log  0.05297  =  2.72403. 

3.  The  Co-Logarithm  (Arithmetical  Complement).  The  co-loga- 
rithm of  a  number  is  the  logarithm  of  that  number  subtracted 
from  unity  or  from  10.     Thus, 

co-log  824  =  10  -  log  824  =  10  -  2.91593  =  7.08407. 

The  CO -logarithm  of  a  number  may  be  written  down  by  beginning 
at  the  left-hand  digit  of  its  logarithm  and  subtracting  each  digit 
from  9  except  the  last  significant  one,  which  must  be  taken  from  10. 


INTRODUCTION  xi 

In  performing  division  by  means  of  logarithms,  the  loga- 
rithm of  the  denominator  must  be  taken  from  that  of  the  nu- 
merator. Instead  of  subtracting  the  logarithm  of  the  denomi- 
nator, its  co-logarithm  may  be  added  to  the  logarithm  of  the 
numerator. 

Example.  log  (564  -  328)  =  log  564  -  log  328 

=  log  564  =  2.75128 

+  co-log  328  =  7.48413 

=  0.23541 

The  common  logarithms  shown  in  Tables  I  and  II  are 
approximations  to  five  decimal  places  of  mantissae.  Table 
III  shows  the  natural  values  of  the  trigonometric  functions 
to  four  decimal  places.  Table  IV  contains  a  miscellaneous 
collection  of  short  tables  each  of  which  is  indicated  by  its 
title. 

In  any  one  of  these  Tables  a  dash  underneath  a  terminal  5, 
thus  5,  indicates  that  the  true  value  is  smaller  than  5. 

Example,  log  7292  =  3.86285  to  five  decimals,  but  from  six- 
place  tables,  log  7292  =  3.862847. 

Table  I 

This  Table,  pages  1-19,  shows  the  common  or  Briggian  loga- 
rithms of  numbers  from  1  to  10,009.  On  page  1  the  character- 
istics and  raantissse  of  the  logarithms  of  numbers  from  1  to  100 
are  shown ;  pages  2-19  show  the  mantisste  only,  the  character- 
istics being  supplied  by  Law  of  the  Characteristic  explained 
above. 

Pages  2-19  show  the  first  three  digits  of  the  number,  whose 
logarithm  may  be  required,  in  column  marked  N ;  the  next  digit 
is  in  a  column  with  the  proper  heading,  0,  1,  2,  •  •  •,  9,  shown  in 
the  horizontal  row  at  top  and  bottom  of  the  page.  If  a  number 
consists  of  more  thaw  four  digits  its  logarithm  may  be  approxi- 
mated by  using  the  column  marked  proportional  parts  (Prop. 
Pts.).  The  first  two  digits  of  the  mantissoe  are  shown  in  column 
marked  0  ;  an  asterisk  *  in  this  Table  indicates  that  the  first  two 
digits  of  the  mantissse  must  be  taken  from  the  following  hori- 
zontal row. 

Examples,     i.    Find  log  4876. 

(i)  Find  first  three  figures  487  in  column  ?f.  p.  9. 

(2)  Follow  alo^  the  horizontal  row  of  487  to  column  6. 

(3)  We  now  have  the  mantissa,  68806. 

(4)  Determine  the  characteristic,  3. 

.-.  log  4876  =  .3.68806. 


xii  INTRODUCTION 

2.  Find  log  54973. 

(1)  Find  the  first  four  figures,  p.  10,  as  above. 

(2)  Determine  the  characteristic,  4. 

log  54970  =  4.74013. 

The  first  two  figures  of  the  mantissa  come  from  line  below. 

(3)  On  the  same  page,     log  54980  =  4.74020, 

showing  an  increase  of  7  in  the  mantissa  for  Id  in  the  number. 

(4)  An  increase  of  3  in  the  number  =  f^  x  7  =  2.1  in  the  mantissa. 

.-.  log  54973  =  4.74013  +  2  =  4.74015. 

The  difference  in  successive  luantisstB  on  page  10  will  be  found 
to  be  7,  8,  or  9 ;  and  the  corrections  for  a  fifth  figure  1,  2,  3,  4,  5, 
6,  7,  8,  9  are  shown  in  the  marginal  column  marked  Prop.  Pts. 

3.  Find  log  34.4764. 

■  We  find  the  first  four  digits  on  p.  6, 

log  34.4700  =  1.53744,  tabular  difference  13, 
correction  6  =  7.8,  from  Prop.  Pts.  column, 

correction  0.4  =  .5, 


.-.  log  34.4764  =  1.. 53752 

4.    Find  JV,  when  log  N=  3.57824. 

(1)  The  characteristic  3  shows  that  N  has  four  digits  to  the  left  of  the 
decimal  point. 

(2)  Find  the  mantissa  nearest  to  57824,  p.  7.  This  is  .57818,  which  is  6 
smaller  than  the  given  mantissa. 

(3)  From  p.  7, 

jV  =  3786  +  corrections  for  6  in  a  tabular  difference  of  12. 
.-.  N  =  3786.5. 

The  Prop.  Pts.  column  shows  this  correction  at  once.  A  little 
practice  will  enable  the  computor  to  make  the  corrections  men- 
tall}^  merely  writing  down  the  corrected  result. 

Table  II 

Table  II,  pages  22-73,  contains  five-place  logarithms  of  the 
trigonometric  sine,  cosine,  tangent,  and  cotangent  for  angles  0* 
to  90°.  In  this  Table  10  has  been  added  throughout  to  each 
characteristic. 

Page  22  shows  the  common  logarithm  of  the  sine  and  tangent 
for  each  second  from  0"  to  3',  and  the  logarithms  of  cosine  and 
cotangent  for  every  second  from  89°  57'  to  90°. 

Pages  23-28  show  the  logarithmic  sine,  tangapt,  and  cosine 
for  every  10"  for  angles  0°  to  2°,  also  the  logarithmic  cosine, 
cotangent,  and  sine  for  every  10"  for  angles  from  88°  to  90°. 
Corrections  for  intermediate  seconds  may  be  easily  interpolated. 


INTRODUCTION  xiii 

Thus, 

log  sin  1°  6'  36"  =  log  sin  1°  6'  30"  + correction  for  6".     See  p.  26. 

log  sin  re' 40"  =  8.28761 
log:  sin  1°  6'  30"  =  8.28652 


x%  of  difference  =  j%  x  (109)  =  6.5.4 
Hence,  log  sin  1'^  6'  36"  =  8.28652  +  65  =  8.28717. 

Pages  29-73  contain  the  logarithms  of  the  sine,  tangent,  co- 
tangent and  cosine  for  each  minute  of  arc  from  0^  to  90°.  The 
degrees  are  indicated  at  the  top  and  bottom  of  each  page,  the 
minutes  are  in  columns  at  the  left  and  right  of  the  page. 

In  columns  marked  d.  are  the  tabular  differences  of  the  log 
sine  and  log  cosine  for  each  minute.  In  column  marked  c.  d. 
are  the  tabular  differences  for  the  log  tangent  (log  cotangent). 
These  tabular  differences  show  the  correction  for  60";  any- 
smaller  number  of  seconds,  for  example,  35",  would  have  |-|  of 
this  tabular  correction.  In  proportional  parts  (Prop.  Pts.)  col- 
umn, these  corrections  are  reduced  to  scale  of  1",  pages  29-34. 
On  pages  35-73  the  tabular  corrections  are  reduced  so  as  to  show 
the  corrections  for  6",  7",  8",  9",  10",  20",  30",  40",  50".  By 
pointing  off  the  corrections  for  10",  20",  30",  40",  50",  corre- 
sponding corrections  for  1",  2",  3",  4",  5"  may  be  obtained. 

The  corrections  for  seconds  should  be  made  mentally  by  aid 
of  the  proportional  parts  column,  and  the  result  of  the  corrected 
logarithm  may  be  written  at  once. 

Examples,     i.    Find  log  sin  28°  37'  21". 

From  p.  57,  log  sin  28°  37'  =  9.68029,  tabular  difference  23. 

Prop.  Pts.  for  20"  =  7.7 

Prop.  Pts.  for    1"  = ^ 

.-.  log  sin  28°  37'  21"  =  9.68037 

2.  Find  log  cos  16°  48'  24". 

On  p.  45,  log  cos  16°  48'  =  9.98106,  tabular  difference  4. 

Pi-op.  Pts.  for  20"  =             1.3    1    ,    ,         ,  .      .  J 
_.    ^   _,     ,      ,„  „„  Mo  be  subtracted. 

Prop.  Pts.  for  4"    =  ^  j 

9.98104 
Correction  for  log  cosines  and  log  cotangents  are  to  be  subtracted. 

3.  Find  the  angle  x,  when  log  tan  x  =  9.65948. 

(1)  Find  in  log  tan  column  a  log  as  near  the  given  one  as  possible,  p.  53. 

(2)  log  tan  24°  32'  =  9.65937, 

with  a  difference  of  1 1  to  a  tabular  difference  of  34. 

(3)  In  Prop.  Pts.  column  11  to  a  scale  of  34  shows  a  20"  correction. 

.-.  X  =  24°  32'  20". 


XIV 


INTRODUCTION 


4.    Find  the  angle  x^  when  log  cot  x  =  9.48726. 

(1)  On  p.  46,  in  log  cot  column, 

log  cot  72°  56'  =  9.48714, 

with  a  difference  of  12  to  a  tabular  difference  of  45. 

(2)  In  Prop.  Pts.  column  45, 

Correction  for  10"  =    7.5 

Correction  for    6"  =    4.5 

Hence,  correction  for  16"  =  12.0 

(3)  Since  cotangents  increase  with  decrease  of  angle,  subtract  the  16"  cor- 
rection, giving  X  =  72°  55'  44". 

In  addition  to  the  acute  angle  set  at  the  top  and  bottom  of 
each  page  in  the  Table,  pages  29-73,  90°  +  the  acute  angle,  180° 
+  the  angle,  and  270°  +  the  angle  are  printed  in  smaller  type. 
Since  the  trigonometric  functions  of  90°+  a;  and  270°+  x  change 
to  co-functions  of  x,  and  the  functions  of  180°  +  x  retain  their 
names,  we  may  find  the  logarithms  of  the  trigonometric  func- 
tions of  any  one  of  these  larger  angles.  In  each  case  the  sign 
of  the  function  must  be  noted  as  shown  in  the  foUowinof  table: 


X 

sin  X 

cosx 

tanx 

cotx 

secx 

cscx 

90°  +  X 

+  cosx 

—  sin  X 

—  cot  X 

—  tan  X 

—  cscx 

-f-  secx 

180°  -f  X 

—  sin  a: 

—  cosx 

+  tan  X 

-\-  cotx 

—  secx 

—  cscx 

270°  +  X 

—  cos  a: 

-f-  sinx 

—  cotx 

—  tanx 

+  CSC  X 

—  sec  X 

In  the  Table  the  angles  marked  by  an  asterisk  (*)  are  made 
up  of  90°+  a;,  or  270°+  x  ;  this  mark  indicates  that  the  co-func- 
tion of  x  is  to  be  taken.  Otherwise  the  direct  function  of  x  is 
to  be  taken,  proper  regard  being  had  for  the  algebraic  sign  in 
each  case. 

Table  III 

Pages  75-93  contain  four-place  tables  of  the  natural  trigono- 
metric functions  for  each  minute.  The  sines  and  cosines  ap- 
pear upon  the  left-hand  pages,  the  tangents  and  cotangents 
upon  the  right. 


Table  IV 

Table  IV,  pages  95-99,  contains  (1)  a  table  of  radian  measure 
0°  to  180°  for  each  degree ;  also  for  1'  to  60'  and  for  V  to  60" ; 
(2)  two  tables  of  natural  logarithms  of  numbers  from  1  to  200 
and  1  to  9.9  by  tenths;  (3)  a  table  of  hyperbolic  functions, 
sinh  X,  cosh  a;,  x  varying  from  0.01  to  1  by  hundredths. 


LOGARITHMIC,    TRIGONOMETEIC, 
AND    OTHER   TABLES 


^''X 


.1^ 


TABLE  I 


y 


COMMON  OR  BRIGGIAN  LOGARITHMS 


OF 


NUMBERS 


From  1  to  10009 

Note.    A  *  in  Table  I  indicates  that  the  first  two  digits  of  the  mantissa  are 

to  be  taken  from  the  following  line. 
A  dash  underneath  a  terminal  6,  thus  5,  indicates  that  the  true  value  is  less  than  5. 


1    100 


log 


log 


log 


log 


log 


8 

9 

10 

11 

12 
13 
14 
15 

16 

17 
18 
19 
20 


!  0.  00  000 
0.  30  103 
0.  47  712 
0.  60  206 
0.  69  897 

0.  77  815 
0.  84  510 
0.90  309 

0.  95  424 

1.  00  000 

1.  04  139 
1.07  918 
1.  11394 
1.  14  613 
1.  17  609 

1.  20  412 
1.  23  045 
1.  25  527 
1.  27  875 
1.  30  103 

log 


21 

22 
23 
24 

25 

26 

27 
28 
29 
30 

31 

32 
33 
34 

35 

36 

37 
38 
39 
40 

N 


1.  32  222 
1.  34  242 
1.  36  173 
1.38  021 
1.  39  794 

1.  41  497 
1.  43  136 
1.  44  716 
1.  46  240 
1.  47  712 

1.  49  136 

1.  50  515 
1.  51  851 
1.  53  148 
1.  54  407 

1.  55  630 
1.  56  820 
1.  57  978 
1.  59  106 
1.  60  206 

log 


41 

42 
43 
44 
45 

46 

47 
48 
49 
50 

51 

52 
53 
54 
55 

66 

57 
58 
59 
60 


1.  61  278 
1.  62  325 
1.  63  347 
1.  64  345 
1.  65  321 

1.  66  276 
1.  67  210 
1.  68  124 
1.  69  020 
1.  69  897 

1.  70  757 
1.  71  600 
1.  72  428 
1.  73  239 
1.  74  036 

1.  74  819 

1.  75  587 
1.  76  343 
1.  77  085 
1.  77  815 

log 


61 

62 
63 
64 
65 

66 

67 
68 
69 
70 

71 

72 
73 
74 
75 

76 

77 
78 
79 
80 


1.  78  533 
1.  79  239 
1.  79  934 
1.  80  618 
1.  81  291 

1.  81  954 
1.  82  607 
1.  83  251 
1.  83  885 
1.  84  510 

1.  85  126 
1.  85  733 
1.  86  332 
1.  86  923 
1.  87  506 

1.  88  081 
1.  88  649 
1.  89  209 
1.  89  763 
1.  90  309 

log 


81 

82 
83 
84 
85 

86 

87 
88 
89 
90 

91 

92 
93 
94 
95 

96 

97 

98 

99 

100 

N 


1.  90  849 
1.  91  381 
1.91908 
1.  92  428 
1.  92  942 

1.  93  450 
1.  93  952 
1.  94  448 
1.  94  939 
1.  95  424 

1.  95  904 
1.  96  379 
1.96  848 
1.  97  313 

1.  97  772 

1.  98  227 
1.  98  677 

1.  99  123 
1.99  564 

2.  00  000 

log 


Logarithms  of  Numbebs 

100-150 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

100 

01 
02 
03 

00  000 
432 
860 

01  284 

043 

087 

130 

173 

217 

260 

689 

*115 

536 

303 

732 

*157 

578 

346 

775 

*199 

620 

389 

817 

*242 

662 

475 
903 
326 

518 
945 
368 

561 
988 
410 

604 

*030 

452 

647 

*072 

494 

I 

44 

4-4 

43 

4-3 

42 

4.2 

04 
05 
06 

703 

02  119 

531 

745 
160 

572 

787 
202 
612 

828 
243 
653 

870 
284 
694 

912 

325 
735 

953 
366 
776 

995 
407 
816 

*036 
449 

857 

*078 
490 
898 

2 

3 
4 
5 

8.8 
13-2 
17.6 

22.0 

8.6 
12.9 
17.2 
21. .5 

8.4 
12.6 
16.8 
21.0 

07 
08 
09 
110 
11 
12 
13 

938 

03  342 

743 

979 
383 

782 

*019 
423 

822 

*060 
463 
862 

*100 
503 
902 

*141 
543 
941 

*181 
583 
981 

*222 

623 

*021 

*262 

663 

*060 

*302 

703 

*100 

6 
7 
8 
9 

26.4 
30.8 
35-2 

39-6 

25.8 
30-1 
34-4 
38.7 

25.2 
29.4 
33.6 
37-8 

04  139 

179 

218 

258 

297 

336 

376 

766 

*154 

538 

415 

454 

844 

*23] 

614 

493 

883 

*269 

652 

1 

532 

922 

05  308 

571 
961 
346 

610 
999 
385 

650 

*038 

423 

689 

*077 

461 

727 

*115 

500 

805 

*192 

576 

I 

2 

41 

4.1 
8.2 

40 

4.0 
8.0 

89 

3.9 
7.8 

14 
15 
16 

690 

06  070 

446 

729 
108 
483 

767 
145 
521 

805 
183 

558 

843 
221 
595 

881 
258 
633 

918 
296 
670 

956 
333 
707 

994 
371 
744 

*032 
408 
781 

3 
4 
S 
6 

12.3 
16.4 
20.S 
24.6 

12.0 
16.0 
20.0 
24.0 

II. 7 
iS-6 
I9-S 
23-4 

17 

18 

19 

120 

21 
22 
23 

819 
07  188 

555 

856 
225 
591 

893 
262 
628 

930 
298 
664 

967 
335 
700 

*004 
372 
737 

*041 
408 
773 

*078 
445 
809 

*115 
482 
846 

*151 
518 
882 

7 
8 
9 

28.7 
32.8 
36.9 

28.0 
32.0 
36.0 

27-3 

31-2 

3S-I 

•   918 

08  279 

636 

991 

954 

314 

672 

*026 

990 

350 

707 

*061 

*027 
386 
743 

*096 

*063 

*099 

*135 

*171 
529 

884 
*237 

*207 

*243 

1 

422 

778 

*132 

458 

814 

*167 

493 

849 

*202 

565 
920 

*272 

600 

955 

*307 

I 

2 

88 

3.8 
7.6 

37 

3-7 
7.4 

86 

3.6 
7.2 

24 
25 
26 

09  342 
691 

10  037 

377 
726 
072 

412 
760 
106 

447 
795 
140 

482 
830 
175 

517 
864 
209 

552 
899 
243 

587 
934 

278 

621 

968 
312 

656 

*003 
346 

3 

4 
S 
6 

11.4 

IS-2 
19.0 
22.8 

II. I 
14.8 
18.S 
22.2 

10.8 
14.4 
18.0 
21.6 

27 
28 
29 
130 
31 
32 
33 

380 
~   721 
11  059 

394 

41i 
755 
093 

449 
789 
126 

483 
823 
160 

517 
857 
193 

551 
890 
227 
561 

585 
924 
261 
594 

619 
958 
294 
628 

653 
992 
327 

687 

*025 

361 

7 
8 
9 

26.6 
30-4 
34.2 

25-9 
29.6 
33-3 

25.2 
28.8 
32.4 

428 

461 

494 

528 

661 

694 

1 

727 

12  057 

385 

760 
090 
418 

793 
123 
450 

826 
156 
483 

860 
189 
516 

893 
222 
548 

926 
254 
581 

959 
287 
613 

992 
320 
646 

*024 
352 
678 

I 

2 

85 

3-S 

7.0 

34 

3-4 
6.8 

33 

3-3 
6.6 

34 
35 
36 

710 

13  033 

354 

743 
066 
386 

775 
098 
418 

808 
130 
450 

840 
162 
481 

872 
194 
513 

905 
226 
545 

937 

258 
577 

969 
290 
609 

*001 
322 
640 

3 
4 
S 
6 

10.5 
14.0 
17-5 
21.0 

10.2 
13-6 
17.0 
20.4 

9.9 

13-2 

16.S 
19.8 

37 
38 
39 
140 
41 
42 
43 

672 

988 

14  301 

704 

*019 

333 

735 

*051 

364 

767 

*082 

395 

799 

*114 

426 

830 

*145 

457 

862 

*176 

489 

893 

*208 

520 

925 
*239 
551 
860 
*168 
473 
776 

956 

*270 
582 

7 
8 
9 

245 
28.0 

315 

23.8 
27.2 
30-6 

23-1 

26.4 

29.7 

613 

922 

15  229 

534 

644 

675 

706 

737 

768 

799 

829 

891 

*198 

503 

806 

1 

953 
259 
564 

983 
290 
594 

*014 
320 
625 

*045 
351 
655 

*076 
381 
685 

*106 
412 
715 

*137 
442 
746 

1 

2 

82 

3-2 

6.4 

31 

31 
6.2 

30 

30 
6.0 

44 
45 
46 

836 

16  137 

435 

866 
167 
465 

897 
197 
495 

927 

227 
524 

957 
256 

554 

987 
286 
584 

*017 
316 
613 

*047 
346 
643 

*077 
376 
673 

*107 
406 
702 

3 
4 
5 
6 

9.6 

12.8 

16.0 

19.2 

9-3 
12.4 
IS-S 
18.6 

9.0 
12.0 
150 
18.0 

47 

48 

49 

150 

732 

17  026 

319 

609 

761 
056 
348 
638 

791 
085 
377 

820 
114 
406 

850 
143 
435 

879 
173 
464 

909 
202 
493 

938 
231 

522 

967 
260 
551 

997 
289 
580 

7 
8 
9 

22.4 
25.6 
28.8 

21.7 
24.8 
27.9 

21.0 
24.0 
27.0 

667 

696 

725 

754 

782 

811 

840 

869 

1 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9  1   Prop.  Pts.  1 

Logarithms  of  Numbers 

150-200 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

150 

51 

17  609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

898 

926  955 

984 

*013 

*041 

*070 

*099 

*127 

*156 

29 

"8 

52 

18  184 

213 

241 

270 

298 

327 

355 

384 

412 

441 

53 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 

I 

2.9 

2.8- 

54 

752 

780 

808 

837 

865 

893 

921 

949 

977 

*005 

3 

8.7 

8.4 

55 

19  033 

061 

089 

117 

145 

173 

201 

229 

257 

285 

4 

II.6 

II. 2 

56 

312 

340 

368 

396 

424 

451 

479 

507 

535 

562 

5 
6 

14-5 
[7.4 

14.0 
16.8 

57 

590 

618 

645 

673 

700 

728 

756 

783 

811 

838 

7 

20.3 

19.6 

58 

866 

893 

921 

948 

976 

*003 

*030 

*0S8 

*085 

*112 

8 

23.2 

22.4 

59 
160 

61 

20  140 

167 

194 

222 

249 

276 

303 

330 

358 

385 

9 

26.1 

25.2 

412 

439 

466 
737 

493 

520 

548 

575 

602 

629 

656 

1 

683 

710 

763 

790 

817 

844 

871 

898 

925 

91 

26 

62 

952 

978 

*005 

*032 

*059 

*085 

*112 

*139 

*165 

*192 

63 

21  219 

245 

272 

299 

325 

352 

378 

40i 

431 

458 

I 

2.7 

2.6 

64 

484 

511 

537 

564 

590 

617 

643 

669 

696 

722 

3 

8.1 

7.8 

65 

748 

775 

801 

827 

854 

880 

906 

932 

958 

985 

4 

10.8 

10.4 

66 

22  011 

037 

063 

089 

115 

141 

167 

194 

220 

246 

5 
6 

13.5 
16.2 

13.0 
1S.6 

67 

272 

298 

324 

350 

376 

401 

427 

453 

479 

505 

7 

18.9 

18.2 

68 

531 

557 

583 

608 

634 

660 

686 

712 

737 

763 

8 

21.6 

20.8 

69 
170 

71 

789 
23  045 

814 

840 

866 

891 

917 
172 

943 

968 

994 

*019 

9 

24.3 

23-4 

070 

096 

121 

147 

198 

223 

249 

274 

1 

300 

325 

350 

376 

401 

426 

452 

477 

502 

528 

25 

72 

553 

578 

603 

629 

654 

679 

704 

729 

754 

779 

73 

80i 

830 

855 

880 

905 

930 

955 

980 

*005 

*030 

1  2-5 

2  S.o 

74 

24  055 

080 

105 

130 

155 

180 

204 

229 

254 

279 

3  7-5 

75 

304 

329 

353 

378 

403 

428 

452 

477 

502 

527 

4  lO.O 

76 

551 

576 

601 

625 

650 

674 

699 

724 

748 

773 

5  12.5 

6  15.0 

77 

797 

822 

846 

871 

895 

920 

944 

969 

993 

*018 

7  17.S 

78 

25  042 

066 

091 

115 

139 

164 

188 

212 

237 

261 

8  20.0 

79 
180 

81 

285 

310 

334 

358 

382 

406 
648 

431 

455 

479 

503 
744 

9  22.5 

527 

551 

575 

600 

624 

672 

696 

720 

1 

768 

792 

816 

840 

864 

888 

912 

935 

959 

983 

24 

23 

82 

26  007 

031 

055 

079 

102 

126 

150 

174 

198 

221 

83 

245 

269 

293 

316 

340 

364 

387 

411 

435 

458 

2 

2.4 
4.8 

2.3 
4.6 

84 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

3 

7.2 

6.9 

85 

717 

741 

764 

788 

811 

834 

858 

881 

905 

928 

4 

9.0 

9.2 

86 

951 

975 

998 

*021 

*045 

*068 

*091 

*114 

*138 

*161 

5 
6 

12.0 

14.4 

ii.S 
13.8 

87 

27  184 

207 

231 

254 

277 

300 

323 

346 

370 

393 

7 

16.8 

16.1 

88 

416 

439 

462 

485 

508 

531 

554 

577 

600 

623 

8 

19.2 

18.4 

89 
190 

91 

646 

669 

692 

715 

738 

761 

784 
*012 

807 

830 

852 
*081 

9 

21.0 

20.7 

875 

898 

921 

944 

967 

989 

*035 

*058 

1 

28  103 

126 

149 

171 

194 

217 

240 

262 

285 

307 

22 

21 

92 

330 

353 

375 

398 

421 

443 

466 

488 

511 

533 

93 

556 

578 

601 

623 

646 

668 

691 

713 

735 

758 

2 

4.4 

4.2 

94 

780 

803 

825 

847 

870 

892 

914 

937 

959 

981 

3 

6.6 

8.8 

6.3 

8.4 

95 

29  003 

026 

048 

070 

092 

115 

137 

159 

181 

203 

96 

226 

248 

270 

292 

314 

336 

358 

380 

403 

42i 

5 
6 

13-2 

lo.s 
12.6 

97 

447 

469 

491 

513 

S3S 

5S7 

579 

601 

623 

645 

7 

1S.4 

14.7 

98 

667 

688 

710 

732 

754 

776 

798 

820 

842 

863 

8 

17.6 

16.8 

99 
200 

885 

907 

929 

951 

973 

994 

*016 

*038 

*060 

*081 

9 

30  103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

N. 

L.  0 

1 

2 

« 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

LOGAKITHMS    OF    NuMBERS 

200-250 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Its. 

200 

01 

30  103 

m 

146 

168 

190 

211 

233 

~ 

276 

298 

320 

341 

363 

384 

406 

428 

449 

471 

492 

514 

o>> 

21 

02 

535 

557 

578 

600 

621 

643 

664 

685 

707 

728 

03 

750 

771 

792 

814 

835 

856 

878 

899 

920 

942 

I 

2.2 
4.4 
6.6 

2.1 

04 

963 

984 

*006 

*027 

*048 

*069 

*091 

*112 

*133 

*154 

3 

6.3 

05 

31  175 

197 

218 

239 

260 

281 

302 

323 

345 

366 

4 

8.8 

8.4 

06 

387 

408 

429 

450 

471 

492 

513 

534 

555 

576 

S 
6 

Il.O 
13-2 

lO-S 
12.6 

07 

597 

618 

639 

660 

681 

702 

723 

744 

765 

785 

7 

IS.4 

14.7 

08 

806 

827 

848 

869 

890 

911 

931 

952 

973 

994 

8 

17.6 

16.8 

09 

210 

11 

32  015 

035 

056 

077 

098 

lis 

139 

160 

181 

201 
408 

9 

19.8 

18.9 

222 

243 

263 

284 

305 

325 

346 

366 

387 

1 

428 

449 

469 

490 

510 

531 

552 

572 

593 

613 

20 

12 

634 

654 

675 

695 

715 

736 

756 

777 

797 

818 

13 

838 

858 

879 

899 

919 

940 

960 

980 

*001 

*021 

I 

2.0 

14 

33  041 

062 

082 

102 

122 

143 

163 

183 

203 

224 

3 

6.0 

15 

244 

264 

284 

304 

325 

345 

365 

385 

405 

425 

4 

8.0 

16 

445 

465 

486 

506 

526 

546 

566 

586 

606 

626 

S 
6 

lO.O 

17 

646 

666 

686 

706 

726 

746 

766 

786 

806 

826 

7 

14.0 

18 

846 

866 

885 

905 

925 

945 

965 

985 

*005 

*025 

8 

16.0 

19 
220 

21 

34  044 
242 

064 

084 

104 

124 

143 

163 
361 

183 
380 

203 
400 

223 
420 
616 

9 

18.0 

262 

282 

301 

321 

341 

1 

439 

459 

479 

498 

518 

537 

557 

577 

596 

19 

22 

635 

655 

674 

694 

713 

733 

753 

772 

792 

811 

23 

830 

850 

869 

889 

908 

928 

947 

967 

986 

*005 

I 

2 

1.9 

3-8 

24 

35  025 

044 

064 

083 

102 

122 

141 

160 

180 

199 

3 

5-7 

25 

218 

238 

257 

276 

295 

315 

334 

353 

372 

392 

4 

7.6 

26 

411 

430 

449 

468 

488 

507 

526 

545 

564 

583 

5 
6 

9-5 

27 

603 

622 

641 

660 

679 

698 

717 

736 

755 

774 

7 

13-3 

28 

793 

813 

832 

851 

870 

889 

908 

927 

946 

965 

8 

15-2 

29 
230 

31 

984 

*003 

*021 

*040 

*059 

*078 

*097 

*116 

*135 

*154 

0 

17. 1 

36  173 

192 

211 

229 

248 

267 

286 

305 

324 

342 
530 

1 

361 

380 

399 

418 

436 

455 

474 

493 

511 

18 

32 

549 

568 

586 

605 

624 

642 

661 

680 

698 

717 

33 

736 

754 

773 

791 

810 

829 

847 

866 

884 

903 

2 

3.6 

34 

922 

940 

959 

977 

996 

*014 

*033 

*051 

*070 

*088 

3 

5-4 

35 

37  107 

125 

144 

162 

181 

199 

218 

236 

254 

273 

4 

7.2 

36 

291 

310 

328 

346 

365 

383 

401 

420 

438 

457 

5 
6 

9.0 
10.8 

37 

475 

493 

511 

530 

548 

566 

585 

603 

621 

639 

7 

12.6 

38 

658 

676 

694 

712 

731 

749 

767 

785 

803 

822 

8 

14.4 

39 
240 

41 

840 

858 

876 

894 

912 

931 

949 

967 

985 

*003 

9 

16.2 

38  021 

039 

057 

075 

093 

112 

130 

148 

166 

184 

1 

202 

220 

238 

256 

274 

292 

310 

328 

346 

364 

17 

42 

382 

399 

417 

435 

453 

471 

489 

507 

525 

543 

43 

561 

578 

596 

614 

632 

650 

668 

686 

703 

721 

2 

3.4 

44 

739 

757 

775 

792 

810 

828 

846 

863 

881 

899 

3 

S  I 
6.8 

45 

917 

934 

952 

970 

987 

*005 

*023 

*041 

*058 

*076 

46 

39  094 

111 

129 

146 

164 

182 

199 

217 

235 

252 

5 
6 

8.S 
10.2 

47 

270 

287 

305 

322 

340 

358 

375 

393 

410 

428 

7 

11.9 

48 

445 

463 

480 

498 

515 

533 

550 

568 

585 

602 

13-6 

49 
250 

620 

637 

655 

672 

690 

707 

724 

742 

759 

777 

9 

iS-3 

794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

Logarithms  of  Numbers 


250-300 


X. 

L.  0 

1  1 

1 

,  2  ,  3 

4 

5   G    7    8   9 

Prop.  Pts. 

250 

51 

39  794 
967 

[  811 
985 

829 
*002 

846 

863 

881  898  915  933  950 

*019 

*037 

*054  *071  *088  ^*106  *123 

18 

52 

40  140 

157 

175 

192 

209 

226  ,  243  i  261  i  278  ,  295 

53 

312 

329 

346 

364 

381 

398 

415 

432  449 

466 

I 

1.8 
7  ft 

54 

483 

500 

518 

535 

552 

569 

586 

603  620 

637 

3 

5-4 

.■)-■) 

654 

671 

688  705 

722 

739  '   756 

773 

790  1  807 

4 

7.2 

56 

824 

841 

858  875 

892 

909  926 

943 

960 

976 

5 
6 

9.0 

I0.8 

57 

993 

*010 

*027  *044 

*061 

*078  *095 

»111 

*128 

*145 

7 

12.6 

58 

41  162 

179 

196  1  212 

229 

246  ;  263  '  280  296 

313 

8 

14.4 

59 
260 

61 

330 

347 

363  1  380 

397 

414  ;  430  447  464 

481 

9 

16.2 

497 

514 

531  1  547 

564 

581  ;  597  614  631 

647 

1 

664 

68L 

697  i  714 

731 

747  764  780  797 

814 

17 

62 

830 

847 

863  1  880 

896 

913  i  929  946 

963 

979 

63 

996 

*012 

*029 

*045 

*062 

*078 

*095  j*lll 

*127 

*144 

1 

1.7 
3-4 
S.I 

64 

42  160 

177 

193 

210 

226 

243 

259 

275 

292 

308 

3 

65 

325 

341 

357 

374 

390 

406 

423 

439 

455 

472 

4 

6.8 

66 

488 

504 

521 

537 

553 

570 

586 

602 

619 

635 

S 
5 

8.S 

67 

651 

667 

684 

700 

716 

732 

749 

765 

781 

797 

7 

11.9 

68 

813 

830 

846  862 

878 

894  1  911 

927 

943 

959 

8 

13-6 

69 

270 

71 

975 

991 

*00S  *024 

*040 

*056 

*072  :*088 

*104 

*120 

9 

15.3 

43  136 
297 

152 
313 

169  185 
329  1  345 

201 

217 

233 

249 

265  :  281 

1 

361 

377 

393 

409 

425  ,  441 

16 

72 

457 

473 

489  505 

521 

537 

553  569 

584 

600 

73 

616 

632 

648 

664 

680 

696 

712  727 

743 

759 

I 

2 

1.6 

3-2 

74 

775 

791 

807 

823 

838 

854 

870  886 

902 

917 

3 

4.8 

/:) 

933 

949 

965  981 

996 

*012 

*028  *044  *059 

*075 

4 

6.4 

76 

44  091 

107 

122 

138 

154 

170 

185 

201  217 

232 

5 
6 

8.0 
9.6 

77 

248 

264 

279 

295 

311 

326 

342 

358  373 

389 

7 

11.2 

78 

404 

420 

436 

451 

467 

483  498 

514  529 

545 

8 

12.8 

79 
280 

81 

560 

576 

592 

607 

623 

638 

654 

669  685 

700 

0 

14.4 

716 

731 

747 

762 

778 

793 

809 

824  840 

855 

1 

871 

886 

902 

917 

932 

948 

963  :  979  994 

*010 

16 

82 

45  025 

040 

056 

071 

086 

102 

117  i  133 

148 

163 

83 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

2 

i-S 

30 

84 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

3 

4-S 

85 

484 

500 

515 

530 

545 

561 

576 

591 

606 

621 

4 

86 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

5 
6 

7-5 
9.0 

87 

788 

803 

818 

834 

849 

864 

879 

894  909 

924 

7 

10.5 

88 

939 

954 

969 

984 

*000 

*015  *030  *045  *060 

*075 

8 

12.0 

89 

290 

91 

46  090 
240 
389 

105 

255 
404 

120 

135 

150 

165 

180 
330 

195  210 

225 

9 

13.S 

270 

285 

300 

315 
464 

345  359 

374 

1 

419 

434 

449 

479  1  494  509 

523 

14 

92 

538 

553 

568 

583 

598 

613  627  ;  642  1  657 

672 

93 

687 

702 

716 

731 

746 

761 

776  790 

805 

820 

2 

1.4 
2.8 

94 

835 

850 

864 

879 

894 

909 

923  938 

953  %7| 

3 

4.2 
S.6 

95 

982 

997 

*012 

*026 

*041 

*056 

*070  *0S5  *100  1*114 1 

4 

% 

47  129 

144 

159 

173 

188 

202 

217  232  .  246 

261 

5 
6 

7.0 
8.4 

97 

276 

290 

305 

319 

334 

349 

363 

378  392 

407 

7 

9.8 

98 

422 

436 

451 

465 

480 

494 

509 

524  538  553 1 

8 

II. 2 

99 
300 

567 

582 

596 

611 

625 

640 

654 

669 

683  698 

9 

712 

727 

741  1  756 

i  770 

784 

799 

813 

828  842 

N. 

L.  0 

1  1 

2 

3 

4 

5   6    7  j  8 

9 

Prop.  Pts. 

LOGABITHMS    OF   NuMBEKS 

300-350 


N. 

L.  0 

1 

2 

3 

4 

o 

6 

7 

8 

9 

Prop.  Pts. 

300 

01 
02 
03 

47  712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

857 

48  001 

144 

871 
015 
159 

885 
029 
173 

900 
044 

187 

914 
058 
202 

929 
073 
216 

943 
087 
230 

958 
101 
244 

972 
116 
259 

986 
130 
273 

15 

04 
05 
06 

287 
430 

572 

302 
444 
586 

316 
458 
601 

330 

473 
61i 

344 
487 
629 

359 
501 
643 

373 
515 
657 

387 
530 
671 

401 

544 
686 

416 

558 
700 

I 

2 
3 

i.S 
3.0 
4-5 

07 
08 
09 
310 
11 
12 
13 

714 
855 
996 

728 

869 

*010 

742 

883 

*024 

756 

897 
*038 

770 

911 

*052 

192 

785 

926 

*066 

206 

799 

940 

*080 

813 

954 
*094 

827 

968 

*108 

841 

982 

*122 

262 

4 
5 
6 
7 
8 
9 

7.S 
g.o 
lo.s 

I2.0 

13-5 

49  136 

150 

164 

178 

220 

234 

248 

276 
415 

554 

290 
429 
568 

304 
443 

582 

318 
457 
596 

332 
471 
610 

346 
485 
624 

360 
499 
638 

374 
513 
651 

388 
527 
665 

402 
541 
679 

14 
15 
16 

693 
831 
969 

707 
845 
982 

721 
859 
996 

734 

872 

*010 

748 

886 

*024 

762 

900 

*037 

776 

914 

*051 

790 

927 

*06i 

803 

941 

*079 

817 

955 

*092 

I 

14 

1.4 

17 

18 

19 

320 

21 
22 
23 

50  106 
243 
379 

120 
256 
393 

133 
270 
406 

147 
284 
420 

161 
297 
433 

174 
311 
447 

188 
325 
461 

202 
338 
474 

215 
352 

488 

229 
365 
501 

2 

3 
4 
5 
6 
7 
8 
9 

2.8 
4.2 

•S.6 
7.0 
8.4 
9.8 

II. 2 
12.6 

515 

529 

542 

556 

569 

583 

596 

610 

623 

637 

651 

786 
920 

664 
799 
934 

678 
813 
947 

691 
826 
961 

705 
840 
974 

718 
853 
987 

732 

866 

*001 

745 

880 

*014 

759 

893 

*028 

772 

907 

*041 

24 
25 
26 

51055 
188 
322 

068 
202 
335 

081 
215 
348 

095 
228 
362 

108 
242 
375 

121 
255 
388 

135 
268 
402 

148 
282 
415 

162 
295 
428 

175 
308 
441 

27 
28 
29 

330 

31 
32 
33 

455 
587 
720 

468 
601 
733 

481 
614 
746 

495 
627 
759 

508 
640 

772 

521 
654 
786 

534 
667 
799 

548 
680 
812 

561 
693 

825 

574 
706 
838 

I 

2 

3 
4 
S 
6 
7 

18 

1-3 

2.6 

3-9 
5.2 
6.S 
7.8 
9-1 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

983 

52  114 

244 

996 

127 
257 

*009 
140 
270 

*022 
153 
284 

*035 
166 
297 

*048 
179 
310 

*061 
192 
323 

*075 
205 
336 

*088 
218 
349 

*101 
231 
362 

34 
35 
36 

375 
504 
634 

388 
517 
647 

401 
530 
660 

414 
543 
673 

427 
556 
686 

440 
569 
699 

453 

582 
711 

466 

595 
724 

479 
608 
737 

492 
621 
750 

8 
9 

10.4 
n.7 

37 

38 

39 

340 

41 

42 
43 

763 

892 

53  020 

776 
905 
033 

789 
917 
046 

802 
930 
058 

815 
943 
071 

827 
956 
084 

840 
969 
097 

853 
982 
110 

866 
994 
122 

879 

*007 

135 

I 

2 

3 
4 

12 

1.3 
2.4 
3.6 
4.8 
6.0 
7.2 
8.4 
9.6 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 

275 
403 
529 

288 
415 
542 

301 
428 

555 

314 

441 
567 

326 
453 
580 

339 
466 
593 

352 
479 
605 

364 
491 
618 

377 
504 
631 

390 
517. 
643 

44 
45 
46 

656 

782 
908 

668 
794 
920 

681 
807 
933 

694 
820 
945 

706 
832 
958 

719 
845 
970 

732 
857 
983 

744 
870 
995 

757 

882 

*008 

769 

895 

*020 

S 
6 
7 
8 

47 

48 

49 

350 

54  033 
158 
283 

045 
170 
295 

058 
183 
307 

070 
195 
320 

083 
208 
332 

095 
220 
345 

108 
233 
357 

120 
245 
370 

133 

258 
382 

145 
270 
394 

9 

407 

419 

432_ 

444 

456 

469 

481 

494 

506 

518 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

Logarithms  of  Numbers 

350-400 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

350 

51 
52 
53 

54  407 

419 

432 

444 

456 
580 
704 
827 

469 
593 
716 
839 

481 
605 
728 
851 

494 

506 

518 

531 
654 

777 

543 
667 
790 

555 
679 
802 

568 
691 
814 

617 

741 
864 

630 
753 

876 

642 

765 
888 

54 
55 
56 

900 

55  023 

145 

913 
035 
157 

925 
047 
169 

937 
060 
182 

949 
072 
194 

962 
084 
206 

974 
096 
218 

986 
108 
230 

998 
121 
242 

*011 
133 
255 

I 

18 

1-3 

57 
58 
59 
360 
61 
62 
63 

267 
388 
509 

279 
400 

522 

291 
413 

534 

303 
425 
546 
666 

315 

437 
558 
678 

328 
449 
570 

340 
461 

582 

352 
473 
594 

364 

485 
606 
727 
847 
967 
*086 

376 
497 
618 
739 
859 
979 
*098 

2 

3 
4 
S 
6 
7 
8 
9 

2.6 

3-9 

S-2 

6.S 
7.8 

Q.I 

I0.4 
II.7 

630 

642 

654 

691 

703 

715 

835 

955 

*074 

751 
871 
991 

763 

883 

*003 

775 

895 

*015 

787 

907 

*027 

799 

919 

*038 

811 

931 

*050 

823 

943 

*062 

64 
65 
66 

56  110 

229 
348 

122 
241 
360 

134 
253 
372 

146 
265 
384 

158 
277 
396 

170 
289 
407 

182 
301 
419 

194 
312 
431 

205 
324 
443 

217 
336 
45i 

67 

68 

69 

370 

71 
72 
73 

467 
585 
703 

478 
597 
714 

490 
608 
726 

502 
620 
738 
855 
972 
089 
206 

514 
632 
750 

526 
644 
761 

538 
656 

773 

549 
667 
785 
902 

561 
679 

797 

573 
691 

808 

I 

2 

3 
4 
S 
6 
7 
8 
9 

1.2 

2.4 

3-6 

4.8 
6.0 

7.2 

8.4 
9.6 
I0.8 

820 

832 

844 

867 

879 

891 

914 

*031 

148 

264 

926 

937 

57  054 

171 

949 
066 
183 

961 
078 
194 

984 
101 
217 

996 
113 
229 

*008 
124 
241 

*019 
136 
252 

*043 
159 
276 

74 
75 
76 

287 
403 
519 

299 
415 
530 

310 
426 

542 

322 
438 
553 

334 

449 
565 

345 
461 
576 

357 

473 
588 

368 
484 
600 

380 
496 
611 

392 
507 
623 

77 

78 

79 

380 

81 
82 
83. 

634 
749 
864 

646 
761 

875 

657 

772 
887 

669 

784 
898 

680 

795 

910 

*024 

692 

807 
921 

703 
818 
933 

715 
830 
944 

726 
841 
955 

738 
852 
967 
*081 
195 
309 
422 

I 
a 
3 
4 
5 
6 
7 

11 

I.I 

2.2 

3-3 
4.4 

li 

7.7 

978 

990 

*001 

*013 

*035 

*047 

*058 

*070 

58  092 
206 
320 

104 
218 
331 

115 
229 
343 

127 
240 
354 

138 
252 
365 

149 
263 
377 

161 

274 
388 

172 
286 
399 

184 
297 
410 

84 
85 
86 

433 
546 
659 

444 
557 
670 

456 
569 
681 

467 
580 
692 

478 
591 
704 

490 
602 
715 

501 
614 
726 

512 
62i 
737 

524 
636 
749 

535 
647 
760 

8 
9 

8.8 
9-9 

87 
88 
89 
390 
91 
92 
93 

771 
883 
995 

782 

894 

*006 

794 

906 

*017 

805 

917 

*028 

816 

928 

*040 

827. 

939 

*051 

162 

838 

950 

*062 

850 
961 

*073 

861 
973 

*084 

872 

984 

*095 

207 

59  106 

118 

129 

140 
251 
362 

472 

151 
262 
373 

483 

173 

184 
295 
406 
517 

195 

I 

3 
3 
4 

10 

I.O 
2.0 

3-0 
4.0 

218 
329 
439 

229 
340 
450 

240 
351 
461 

273 

384 
494 

284 
395 
506 

306 
417 

528 

318 
428 
539 

94 
95 
96 

550 
660 
770 

561 
671 

780 

572 
682 
791 

583 
693 
802 

594 
704 
813 

605 
715 
824 

616 

726 
835 

627 
737 
846 

638 

748 
857 

649 

759 
868 

S 
6 
7 
8 
9 

6.0 

1% 
9.0 

97 

98 

99 

400 

879 

988 

60  097 

890 
999 
108 

901 

*010 

119 

912 

*021 

130 

239 

923 

*032 

141 

934 

*043 

152 

945 

*054 

163 

271 

956 

*065 

173 

282 

966 

*076 

184 

977 

*086 

195 

304 

206 

217 

228 

249 

260 

293 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

Logarithms  of  Numbers 

400-450 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

400 

01 
02 
03 

60  206 

217 

228 

239 

347 
455 
563 

249 

260 

271 

282 

293 

509 
617 

304 

314 
423 
531 

325 
433 
541 

336 

444 
552 

358 
466 
574 

369 

477 
584 

379 

487 
595 

390 
498 
606 

412 
520 
627 

04 
05 
06 

638 
746 
853 

649 
756 
863 

660 
767 

874 

670 

778 
885 

681 

788 
895 

692 
799 
906 

703 
810 
917 

713 
821 
927 

724 
831 
938 

735 
842 
949 

07 

08 

09 

410 

11 
12 
13 

959 

61  066 

172 

970 
077 
183 

981 
087 
19+ 

991 
098 
20+ 

*002 
109 
215 

*013 
119 

225 

*023 
130 
236 

*034 
140 

247 

*045 
151 

257 

*0S5 
162 
268 

I 

2 

3 
4 
S 
6 
7 
8 

11 

I.I 

2.2 
3-3 
4-4 
S-S 
6.6 
7.7 
8.8 

278 

289 

300 

310 

321 

331 

342 

352 

363  374 1 

384 
490 
595 

395 
500 
606 

405 
511 
616 

416 
521 
627 

426 
532 
637 

437 
542 
648 

448 
553 
658 

458 
563 
669 

469 
574 
679 

479 
584 
690 

14 
15 
16 

700 
805 
909 

711 
815 
920 

721 
826 
930 

731 
836 
9+1 

742 
847 
951 

752 
857 
962 

763 
868 
972 

773 
878 
982 

784 
888 
993 

794 

899 

*003 

9 

9.9 

17 

18 

19 

420 

21 
22 
23 

62  014 
118 
221 

024 

128 
232 

034 
138 
242 

045 
149 

252 

055 
159 
263 

066 
170 
273 

076 
180 
284 
387 
490 
593 
696 

086 
190 
294 

397 
500 
603 
706 

097 
201 
304 

107 
211 
315 

325 

335 

346 

356 

366 

377 

408 

418 

428 
531 
634 

439 
542 
644 

449 

552 
655 

459 
562 
665 

469 

572 
675 

480 
583 
685 

511 
613 
716 

521 
624 
726 

10 

24 

25 
26 

737 
839 
941 

747 
849 
951 

757 
859 
961 

767 
870 
972 

778 
880 
982 

788 
890 
992 

798 

900 

*002 

808 

910 

*012 

818 

921 

*022 

829 

931 

*033 

2 
3 
4 
5 
6 

2.0 

3-0 
4.0 
5.0 
6.0 

27 
28 
29 
430 
31 
32 
33 

63  043 
144 
246 

053 
155 
256 

063 
165 
266 

073 
175 
276 

083 
185 
286 

094 
195 
296 

104 
205 
306 

114 
215 
317 
417 

124 

225 
327 

134 
236 
337 

7 
8 
9 

7.0 
8.0 
9.0 

347 

357  i  367 

377 

387 
488 
589 
689 

397 

407 

428 

438 

448 
548 
649 

458  1  468 
558  i  568 
659  i  669 

478 
579 
679 

498 
599 
699 

508 
609 
709 

518 
619 
719 

528 
629 
729 

538 
639 
739 

34 
35 
36 

749 
849 
949 

759 

859 
959 

769 
869 
969 

779 
879 
979 

789 
889 
988 

799 
899 
998 

809 

909 

*008 

819 

919 

*018 

829 

929 

*028 

839 

939 

*038 

q 

37 

38 

39 

440 

41 
42 
43 

64  048 
147 
246 

058 
157 
256 

068 
167 
266 
365 

078 
177 
276 

088 
187 
286 

098 
197 
296 

108 
207 
306 
404 

118 
217 
316 

128 
227 
326 

137 
237 
335 

I 

2 

3 
4 
S 
6 
7 
8 
9 

0.9 

1.8 
2.7 
3.6 
4-5 
5-4 
6.3 
7.2 
8.1 

345 

355 

375 

385 

395 

414 

424 

434 

444 
542 
640 

454 
552 
650 

464 
562 
660 

473 
572 
670 

483 
582 
680 

493 
591 
689 

503 
601 
699 

513 
611 
709 

523 
621 
719 

532 
631 
729 

44 
45 
46 

738 
836 
933 

748 
846 
943 

758 
856 
953 

768. 
865 
963 

777 
875 
972 

787 
885 
982 

797 
895 
992 

807 

904 

*002 

816 

914 

*011 

826 

924 

*021 

47 

48 

49 

450 

65  031 
128 

225 

040 
137 
234 

050 

147 
244 

060 
157 

254 

070 
167 
263 

079 
176 
273 

089 
186 
283 

099 
196 

292 

108 
205 
302 

118 
215 
312 

321 

331 

341 

350 

360 

369 

379 

389 

398 

408 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9    Prop.  Pts.  1 

Logarithms  of  Numbers 

450-500 


x\. 

L.  0 

1 

2 

3 

4 

o 

6    7 

8 

9 

Prop.  Pts. 

450 

51 
52 
53 

65  321 

331 

341 

350  1  360 

369 

379  389 

398 

408 

418 
514 
610 

.427 
523 
619 

437 
533 
629 

447  456 
543  552 

639  648 

1 

466 
562 
658 

475 
571 
667 

485 
581 
677 

495 
591 
686 

504 
600 
696 

54 
55 
56 

706 
801 
896 

715 
811 
906 

725 
820 
916 

734 
830 
925 

744 
839 
935 

753 
849 
944 

763 
858 
954 

772 
868 
963 

782 
877 
973 

792 
887 
982 

57 
58 
59 
460 
61 
62 
63 

992 

66  087 

181 

*001 
096 
191 

*011 
106 

200 

*020 
115 
210 

*030 
124 
219 

*039 
134 

229 

*049 
143 

238 

*058 
153 
247 

*068 
162 

257 

*077 
172 
266 

I 

2 

3 
4 
5 
6 
7 
8 

10 

I.O 
2.0 

3.0 
4.0 
S-o 
6.0 
7.0 
8.0 

276 

285 

295 

304 

314 

323 

332 

342 

351 

361 

370 
464 

558 

380 
474 
567 

389 
483 
577 

398 
492 
586 

408 
502 
596 

417 
511 
605 

427 
521 
614 

436 
530 
624 

445 
539 
633 

455 
549 
642 

64 
65 
66 

652 

745 
839 

661 

755 
848 

671 

764 
857 

680 
773 
867 

689 
783 
876 

699 

792 
885 

708 
801 
894 

717 
811 
904 

727 
820 
913 

736 
829 
922 

9 

9.0 

67 

68 

69 

470 

71 

72 
73 

932 

67  025 

117 

941 
034 
127 

950 
043 
136 

960 
052 
145 

969 
062 
154 

978 
071 
164 

987 
080 
173 

997 
0S9 
182 

*006  *015 
099  108 
191  201 

210 

219  i  228 

237 

247 

256 

265 

274 

284  293 

302 

394 
486 

311 

403 
495 

321 
413 
504 

330 
422 
514 

339 
431 
523 

348 
440 
532 

357 
449 
541 

367 
459 
550 

376 
468 
560 

385 
477 
569 

X 

9 

0.9 

74 
75 
76 

578 
669 
761 

587 
679 
770 

596 
633 
779 

605 
697 
788 

614 
706 
797 

624 
715 
806 

633 

724 
815 

642 
733 

825 

651 
742 
834 

660 

752 
843 

2 

3 
4 
5 
6 

1.8 
2.7 
3.6 
4-5 
S-4 

77 
78 
79 
480 
81 
82 
83 

852 

943 

68  034 

124 

861 
952 
043 

870 
961 
052 

879 
970 
061 

888 
979 
070 

897 
988 
079 
169 

906 
997 

088 

916 

*006 

097 

925 

*015 

106 

934 
«024 
115 
205 
296 
386 
476 

7 
8 
9 

6.3 
7.2 
8.1 

133 

142 

151 

160 

178 

187 

196 

215 
305 
39| 

224 
314 
404 

233 
323 
413 

242 
332 
422 

251 
341 
431 

260 
350 
440 

269 
359 
449 

278 
368 
458 

287 
377 
467 

84 
85 
86 

485 
574 
664 

494 
583 
673 

502 
592 
681 

511 
601 
690 

520 
610 
699 

529 
619 

708 

538 
628 

717 

547 
637 
726 

556 
646 
735 

565 
655 
744 

g 

87 
88 
89 
490 
91 
92 
93 

753 
842 
931 

762 
851 
940 

771 
860 
949 

780 
869 
958 

789 
878 
966 

797 
886 
975 

806 
895 
984 

815 
904 
993 

824 

913 

*002 

833 

922 

*011 

I 

2 

3 
4 
5 
6 
7 
8 
9 

0.8 
1.6 

2.4 

3-2 

4.8 

6.4 
7.2 

69  020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

108 
197 

285 

117 
205 
294 

126 
214 
302 

135 
223 
311 

144 
232 
320 

152 
241 
329 

161 
249 
338 

170 
258 
346 

179 
267 
355 

188 
276 
364 

94 
95 
96 

97 

98 

99 

500 

373 

461 
548 

636 
723 
810 

897 

381 
469 

557 

644 
732 
819 

390 
478 
566 

653 
740 

827 

399 

487 
574 

662 
749 
836 

408 
496 
583 

417 

504 
592 

425 
513 
601 

688 

775 
862 

434 
522 
609 

697 

784 
871 

443 
531 
618 

705 
793 
880 

452 
539 
627 

714 
801 

888 

671 

758 
845 

679 

767 
854 

906 

914 

923 

932 

940 

949 

958 

966 

975 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

Logarithms  of  Numbers 

500-550 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts, 

500 

01 
02 
03 

69  897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

984 

70  070 

157 

992 
079 
165 

»001 
088 
174 

*010 
096 
183 

*018 
105 
191 

*027 
114 
200 

*036 
122 
209 

*044 
131 
217 

*053* 
140 
226 

*062 
148 
234 

04 
05 
06 

243 
329 
415 

252 
338 
424 

260 
346 
432 

269 
355 
441 

278 
364 
449 

286 
372 
458 

295 
381 
467 

303 
389 
475 

312 
398 
484 

321 
406 
492 

07 

08 

09 

610 

11 

12 
13 

501 
586 
672 

509 
595 
680 

518 
603 
689 

526 
612 
697 

535 
621 
706 

544 
629 

714 

552 
638 
723 

561 
646 
731 

569 

655 
740 

578 
663 
749 
834 

I 

2 

3 
4 
5 
6 
7 
8 

9 

O.Q 

1.8 

2.7 

3.6 

4-S 
5.4 
6.3 

7.2 

757 

766 

774 

783 

791 

800 

808 
893 
978 
063 

817 

825 

842 

927 

71  012 

851 
935 
020 

859 
944 
029 

868 
952 
037 

876 
961 
046 

885 
969 
054 

902 
986 
071 

910 
995 
079 

919 

*003 

088 

14 
15 
16 

0% 

181 
265 

105 
189 
273 

113 
198 

282 

122 
206 
290 

130 
214 
299 

139 
223 
307 

147 
231 
315 

155 
240 
324 

164 
248 
332 

172 
257 
341 

9  8.1 

17 

18 

19 

520 

21 
22 
23 

349 
433 
517 

357 
441 

525 

366 
450 
533 

374 
458 
542 

383 
466 
550 

391 

475 
559 
642 

399 
483 
567 
650 

408 
492 

575 

416 
500 
584 

425 
508 
592 

600 

609  1  617 

625 
709 
792 

875 

634 
717 
800 
883 

659 

667 

675 

684 
767 

850 

692 

775 
858 

700 
784 
867 

725  ■   734 
809  :  817 
892  900 

742 
825 
908 

750 
834 
917 

759 
842 
925 

8 

0.8 

24 
25 
26 

933 

72  016 

099 

941 
024 
107 

950 
032 
115 

958 
041 
123 

966 
049 
132 

975 
057 
140 

983 
066 
148 

991 
074 
156 

999 
082 
165 

*008 
090 
173 

2 

3 
4 
5 
6 

1.6 

2.4 

3.2 
4.0 
4.8 

27 
28 
29 
530 
31 
32 
33 

181 
263 
346 

189 

272 
354 

198 
280 
362 
444 

206 
288 
370 

214 
296 
378 

111 
304 
387 

230 

313 
395 
477 

239 

321 
403 

247 
329 
411 

255 
337 
419 

7 
8 
9 

5.6 
6.4 

7.3 

428 

436 

452 

460 

469 

485 

493 

501 

509 
591 
673 

518 
599 
681 

526 
607 
689 

534 
616 
697 

542 
624 
705 

550 
632 
713 

558 
640 
722 

567 
648 
730 

575 
656 
738 

583 
665 
746 

34 
35 
36 

754 
835 
916 

762 
843 
925 

770 
852 
933 

779 
860 
941 

787 
868 
949 

795 
876 
957 

803 
884 
%5 

811 
892 
973 

819 
900 
981 

827 
908 
989 

17 

37 
38 
39 
540 
41 
42 
43 

997 

73  078 

159 

*006 
086 
167 

»014 
094 
175 

*022 
102 
183 

*030 
111 
191 

*038 
119 
199 

*046 
127 
207 

*054 
135 
215 

«062 
143 
223 
304 

*070 
151 
231 

I 

2 

3 
4 
S 
6 
7 
8 
9 

0.7 
1.4 

2.1 
2.8 

3-5 
4.2 
4.9 
5.6 
6.3 

239 

247 

255 

263 

272 

280 

288 

296 

312 

320 
400 
480 

328 
408 
488 

336 
416 
496 

344 
424 
504 

352 
432 
512 

360 
440 
520 

368 
448 
528 

376 
456 
536 

384 
464 
544 

392 
472 

552 

44 
45 
46 

560 
640 
719 

568 
648 
727 

576, 
6561 
735 

584 
664 
^43 

592 
672 
751 

600 
679 
759 

608 
687 
767 

616 
695 

775 

624 
703 
783 

632 
711 
791 

47 

48 

49 

550 

799 
878 
957 

807 
886 
%5 

815 
894 
973 

8^ 
902 
981 

830 
910 
989 

838 
918 
997 

846 

926 

*005 

854 

933 

*013 

862 

941 

*020 

870 
949 

*028 
107 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

10 


Logarithms  of  Numbers 

550-600 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

550 

51 

52 
53 

74  036 
115 
194 
273 

044 

052 

060 

068 

076 

084 

092 

099 

107 

123 

202 
280 

131 
210 

288 

139 
218 
296 

147 
225 
304 

155 
233 
312 

162 
241 
320 

170 
249 
327 

178 

257 
335 

186 
265 
343 

54 
55 
56 

351 
429 
507 

359 

437 
515 

367 

445 
523 

374 
453 
531 

382 
461 
539 

390 

468 
547 

398 

476 
554 

406 

484 
562 

414 
492 
570 

421 
500 

578 

57 
58 
59 
560 
61 
62 
63 

586 
663 
741 

593 
671 
749 

601 
679 

757 

609 
687 
764 

617 
695 

772 

624 

702 
780 

632 
710 

788 

640 

718 
796 

648 
726 
803 

656 
733 
811 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

896 

974 

75  051 

904 
981 
059 

912 
989 
066 

920 
997 
074 

927 

*005 

082 

935 

*012 

089 

943 

*020 

097 

950 

*028 

105 

958 

*035 

113 

966 

*043 

120 

8 

64 
65 
66 

128 
205 
282 

136 
213 

289 

143 
220 
297 

151 
228 
305 

159 
236 
312 

166 
243 
320 

174 
251 
328 

182 
259 
335 

189 
266 
343 

197 
274 
351 

I 

2 
3 
•  4 
S 
6 
7 
8 
9 

0.8 
1.6 

2.4 
3-2 

4-? 

4.8 
6.4 

7.2 

67 

68 

69 

570 

71 

72 
73 

358 
435 
511 

366 

442 
519 

374 
450 
526 

381 

458 
534 

389 
465 
542 

397 
473 
549 

404 
481 

557 

412 
488 
565 

420 
496 

572 
648 

427 
504 
580 

587 

595 

603 

610 

618 

626 

633 

641 

656 

664 
740 

815 

671 

747 
823 

679 

755 
831 

686 
762 
838 

694 
770 
846 

702 

778 
853 

709 

785 
861 

717 
793 
868 

724 
800 
876 

732 
808 
884 

74 
75 
76 

891 

967 

76  042 

899 
974 
050 

906 

982 
057 

914 
989 
065 

921 
997 
072 

929 

*005 

080 

937 

*012 
087 

944 

*020 

095 

952 

*027 

103 

959 

*035 

110 

77 
78 
79 
680 
81 
82 
83 

118 
193 
268 

125 
200 

275 

133 

208 
283 
358 
433 
507 
582 

140 
215 
290 
365 
440 
515 
589 

148 
223 
298 

155 
230 
305 

163 
238 
313 

170 
245 
320 

178 
253 
328 

185 
260 
335 
410 

343 

350 

373 

380 

388 

395 

403 

477 
552 
626 

418 
492 
567 

425 
500 

574 

448 
522 
597 

455 
530 
604 

462 
537 
612 

470 
545 
619 

485 
559 
634 

I 

7 

0.7 

84 
85 
86 

641 
716 
790 

649 
723 
797 

656 
730 
805 

664 
738 
812 

671 
745 
819 

678 

753 
827 

686 
760 
834 

693 

768 
842 

701 

775 
849 

708 
782 
856 

3 
4 
S 
6 
7 

2.1 
2.8 

3-5 
4.2 
il.O 

87 
88 
89 
590 
91 
92 
93 

864 

938 

77  012 

871 
945 
019 

879 
953 
026 
100 

886 
960 
034 

893 
967 
041 

901 
975 
048 
122 

908 
982 
056 

916 
989 
063 
137 

923 
997 
070 

930 

*004 

078 

8 
9 

5.6 
6.3 

085 

093 

107 

115 

129 
203 
276 
349 

144 

151 

159 
232 
305 

166 
240 
313 

173 
247 
320 

181 
254 
327 

188 
262 
335 

195 
269 
342 

210 
283 
3-57 

217 
291 
364 

225 
298 
371 

94 
95 
96 

379 

452 
525 

386 
459 
532 

393 
466 
539 

401 
474 
546 

408 
481 
554 

415 
488 
561 

422 
495 
568 

430 
503 
576 

437 
510 
583 

444 
517 
590 

97 

98 

99 

600 

597 
670 
743 

605 
677 
750 

612 

685 
757 

619 
692 
764 

627 
699 

772 

634 
706 

779 

641 
714 

786 

648 
721 
793 

656 
728 
801 

663 
735 
808 

815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

11 


Logarithms  of  Numbers 

600-650 


X. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

600 

01 
02 
03 

77  815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

887 

960 

78  032 

895 
967 
039 

902 
974 
046 

909 
981 
053 

916 
988 
061 

924 
996 
068 

931 

*003 

07i 

938 

*010 

082 

945 

*017 

089 

952 

*025 

097 

04 
05 
06 

104 
176 

247 

111 
183 
254 

118 
190 
262 

125 
197 
269 

132 
204 
276 

140 

211 
283 

147 
219 
290 

154 
226 
297 

161 
233 
305 

168 
240 
312 

07 

08 

09 

610 

11 

12 
13 

319 
390 

462 

326 
398 
469 

333 
405 
476 

340 
412 

483 

347 
419 
490 

355 
426 
497 

362 

433 
504 

369 
440 
512 

376 
447 
519 

383 
455 
526 

I 

2 

3 
4 
S 
6 
7 
8 

8 

0.8 
1.6 

2.4 

3-2 
4.0 
4.8 
S.6 
6.4 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

604 
675 
746 

611 

682 
753 

618 
689 
760 

625 
696 
767 

633 

704 
774 

640 
711 

781 

647 
718 
789 

654 

725 
796 

661 
732 
803 

668 
739 
810 

14 
15 
16 

817 
888 
958 

824 
895 
965 

831 
902 
972 

838 
909 
979 

845 
916 
986 

852 
923 
993 

859 

930 

*000 

866 

937 

*007 

873 

944 

*014 

880 

951 

*021 

9  /••' 

17 
18 
19 
620 
21 
22 
23 

79  029 
099 
169 

036 
106 
176 

043 
113 
183 

050 
120 
190 

057 
127 
197 

064 
134 
204 

071 
141 
211 

078 
148 
218 

085 
155 
225 

092 
162 
232 

239 

246 

253 

260 

267 

274 
344 
414 
484 

281 
351 
421 
491 

288 

295 

302 

309 
379 
449 

316 
386 
456 

323 
393 
463 

330 
400 
470 

337 
407 
477 

358 
428 
498 

365 
435 
505 

372 
442 
511 

I 

7 
0.7 

24 
25 
26 

518 
588 
657 

525 
595 
664 

532 
602 
671 

539 

609 
678 

546 
616 
685 

553 
623 
692 

560 
630 
699 

567 
637 
706 

574 
644 
713 

581 
650 
720 

3 

3 
4 
S 
6 

1-4 
2.1 
2.8 

3-5 

4.2 

27 
28 
29 
630 
31 
32 
33 

727 
796 
865 
934 
80  003 
072 
140 

734 

803 
872 

741 
810 
879 

748 
817 
886 

754 
824 
893 

761 

831 
900 

768 
837 
906 

775 
844 
913 

782 
851 
920 

789 
858 
927 

7 
8 
9 

4-9 
5.6 
6.3 

941 

948 

955 

962 

969 

975 

982 

989 
058 
127 
195 

996 

010 
079 

147 

017 
085 
154 

024 
092 
161 

030 
099 
168 

037 
106 
175 

044 
113 
182 

051 
120 
188 

065 
134 
202 

34 
35 
36 

209 
277 
346 

216 
284 
353 

223 
291 
359 

229 
298 
366 

236 
305 
373 

243 
312 
380 

250 
318 
387 

257 
325 
393 

264 
332 
400 

271 
339 
407 

6 

37 
38 
39 

640 

41 
42 
43 

414 

482 
550 

421 
489 

557 

428 
496 
564 

434 
502 
570 

441 
509 

577 

448 
516 
584 

455 
523 
591 

462 
530 
598 

468 
536 
604 

475 
543 
611 

I 

2 
3 
4 
5 
6 
7 
8 
9 

0.6 

1.2 

1.8 

2.4 

3.0 
3.6 
4.2 
4.8 
5-4 

618 

625 

632 

638 
706 
774 
841 

645 

652 

659 

665 

672 

679 

747 
814 
882 

686 
754 
821 

693 
760 
828 

699 
767 
835 

713 

781 
848 

720 

787 
855 

726 

794 
862 

733 
801 
868 

740 
808 
875 

44 
45 
46 

889 

956 

81023 

895 
963 
030 

902 
969 
037 

909 
976 
043 

916 
983 
050 

922 
990 
057 

929 
996 
064 

936 

*003 

070 

943 

*010 

077 

949 

*017 

084 

47 

48 

49 

650 

090 

158 
224 

097 
164 
231 

104 
171 
238 

111 

178 
245 

117 
184 
251 

124 
191 
258 

131 
198 
265 

137 
204 
271 

144 
211 

278 

151 
218 

285 

291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

12 


Logarithms  of  Numbers 

650-700 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

650 

51 
52 
53 

81  291 

298 

305 

311 

378 
445 
511 

318 

325 

331 
398 
465 
531 

338 
405 
47i 
538 

345 
411" 

478 
544 

351 
418 
485 
551 

358 
425 
491 

365 
431 
498 

371 
438 
505 

385 
451 
518 

391 

458 
525 

54 
55 
56 

558 
624 
690 

564 
631 
697 

571 
637 
704 

578 
644 
710 

584 
651 
717 

591 
657 
723 

598 
664 
730 

604 
671 

737 

611 
677 
743 

617 
684 
750 

57 
58 
59 
660 
61 
62 
63 

757 
823 
889 
954 

763 
829 

895 

770 
836 
902 

776 
842 
908 

783 
849 
915 

790 
856 
921 
987 

796 
862 
928 
994 
060 
125 
19a 

803 

869 

935 

*000 

809 
875 
941 

816 

882 
948 

961 

968 

974 
040 
105 
171 

981 

*007 

*014 

82  020 
086 
151 

027 
092 

158 

033 
099 
164 

046 
112 
178 

053 
119 
184 

066 
132 
197 

073 
138 
204 

079 
145 
210 

1 

7 

0.7 

64 
65 
66 

217 
282 
347 

223 
289 
354 

230 
295 
360 

236 
302 
367 

243 
308 
373 

249 
315 
380 

256 
321 
387 

263 
328 
393 

269 
334 
400 

276 
341 
406 

2 

3 
4 
S 
6 

1.4 
2.1 
2.8 

3-5 
4.2 

67 

68 

69 

670 

71 
72 
73 

413 
478 
543 

419 
484 
549 

426 

491 
556 

432 
497 
562 

439 
504 
569 

445 
510 

575 

452 
517 

582 

458 
523 
588 

465 
530 
595 

471 
536 
601 

i 

9 

4.9 

|-^ 
6.3 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

672 
737 
802 

679 
743 
808 

685 
750 
814 

692 
756 
821 

698 
763 
827 

705 
769 
834 

711 
776 
840 

718 
782 
847 

724 
789 
853 

730 
795 
860 

74 
75 
76 

866 
930 
995 

872 

937 

*001 

879 
943 

*008 

885 

950 

*014 

892 

956 

*020 

898 

963 

*027 

905 

969 

*033 

911 

975 

*040 

918 

982 
*046 

924 

988 

*052 

77 

78 

79 

680 

81 
82 
83 

83  059 
123 
187 

065 
129 
193 

072 
136 
200 

078 
142 
206 

085 
149 
213 
276 

091 
155 
219 
283 

097 
161 

225 

104 
168 
232 

110 
174 

238 

117 
181 
245 
308 

251 

257 

264 

270 

289 
353 
417 
480 

2% 

302 

315 
378 
442 

321 
385 
448 

327 
391 
45i 

334 
398 
461 

340 
404 
467 

347 
410 
474 

359 
423 
487 

366 
429 
493 

372 
436 
499 

1 

6 

0.6 

84 
85 
86 

506 
569 
632 

512 
575 
639 

518 
582 
645 

525 
588 
651 

531 
594 
658 

537 
601 
664 

544 
607 
670 

550 
613 
677 

556 
620 
683 

563 
626 
689 

2 

3 
4 
S 
6 

1.2 

1.8 

2.4 

3.0 
3.6 

87 
88 
89 
690 
91 
92 
93 

696 

759 
822 

702 

765 
828 

708 
771 
835 

715 
778 
841 

721 

784 
847 

727 
790 
853 

734 
797 
860 

740 
803 
866 

746 
809 

872 

753 
816 
879 

7 
8 
9 

4.2 

4.8 
5.4 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

948 

84  011 

073 

954 
017 
080 

960 
023 
086 

967 
029 
092 

973 
036 
098 

979 
042 
105 

985 
048 
111 

992 
055 
117 

998 
061 
123 

*004 
067 
130 

94 
95 
96 

136 
198 
261 

142 
205 
267 

148 
211 
273 

155 
217 
280 

161 

223 
286 

167 
230 
292 

173 
236 
298 

180 
242 
305 

186 
248 
311 

192 

255 
317 

97 

98 

99 

700 

323 
386 

448 

330 
392 
454 

336 

398 
460 

522 

342 
404 
466 

348 
410 
473 

354 
417 
479 

361 
423 
485 

547 

367 
429 
491 

373 
435 
497 

379 
442 
504 

510 

516 

528 

535 

541 

553 

559 

566 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

13 


LOGAKITHMS    OF   NuMBEBS 

700-750 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

700 

01 
02 
03 

84  510 

516 

522 

528 

535 

541 

547 

553 

559 
621 
683 
745 

566 

572 
634 
696 

578 
640 
702 

584 
646 
708 

590 
652 
714 

597 
658 
720 

603 
665 
726 

609 
671 
733 

615 
677 
739 

628 
689 
751 

04 
05 
06 

757 
819 
880 

763 

825 
887 

770 
831 
893 

776 
837 
899 

782 
844 
905 

788 
850 
911 

794 
856 
917 

800 
862 
924 

807 
868 
930 

813 
874 
936 

07 
08 
09 
710 
11 
12 
13 

942 

85  003 

065 

948 
009 
071 

954 
016 

077 

960 
022 
083 

%7 
028 
089 

973 
034 
095 

979 
040 
101 

985 
046 
107 

991 
052 
114 

997 
058 
120 

I 

2 

3 
4 
5 
6 
7 
8 

7 

0.7 
1.4 

2.1 
2.8 

3.S 
4-2 
4.9 
S.6 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

187 
248 
309 

193 
254 
315 

199 
260 
321 

205 
266 
327 

211 
272 
333 

217 
278 
339 

224 
285 
345 

230 
291 

352 

236 
297 

358 

242 
303 
364 

14 
15 
16 

370 
431 
491 

376 

437 
497 

382 
443 
503 

388 
449 
509 

394 

455 
516 

400 
461 

522 

406 

467 
528 

412 
473 

534 

418 
479 
540 

425 
485 
546 

9 

6.3 

17 

18 

19 

720 

21 
22 
23 

552 
612 
673 

558 
618 
679 

564 
625 
685 

570 
631 
691 

576 
637 
697 

757 

582 
643 
703 
763 

588 
649 
709 
769 

594 
655 
715 

600 
661 
721 

606 
667 

727 

733 

739 

745 

751 

775 

781 

788 

794 
854 
914 

800 
860 
920 

806 
866 
926 

812 
872 
932 

818 
878 
938 

824 
884 
944 

830 
890 
950 

836 
896 
956 

842 
902 
962 

848 
908 
968 

6 

o6 

24 
25 
26 

974 

86  034 

094 

980 
040 
100 

986 
046 
106 

992 
052 
112 

998 
058 
118 

*004 
064 
124 

*010 
070 
130 

*016 
076 
136 

*022 
082 
141 

*028 
088 
147 

I 
4 

1 

1.2 

1.8 

2.4 

3.0 
3-6 

27 
28 
29 
730 
31 
32 
33 

153 
213 
273 

159 
219 
279 

165 

225 
285 

171 
231 
291 

177 
237 
297 

183 
243 
303 

189 
249 
308 

195 

255 
314 

201 
261 
320 

207 
267 
326 

I 

4.8 
S-4 

332 

338 

344 

350 

356 

362 

368 

374 

380 

386 

392 
451 
510 

398 
457 
516 

404 
463 
522 

410 
469 

528 

415 
475 
534 

421 
481 
540 

427 
487 
546 

433 
493 

552 

439 
499 

558 

445 
504 
564 

34 
35 
36 

570 
629 
688 

576 
635 
694 

581 
641 
700 

587 
646 
705 

593 
652 
711 

599 
658 
717 

605 
664 
723 

611 
670 
729 

617 
676 
735 

623 
682 
741 

37 
38 
39 
740 
41 
42 
43 

747 
806 
864 

753 
812 
870 

759 
817 
876 

764 

823 
882 

770 
829 

888 

776 
835 
894 

782 
841 
900 

788 
847 
906 

794 
853 
911 

800 
859 
917 

1 

2 

3 
4 
5 
6 
7 
8 
9 

o.S 

I.O 

i.S 

2.0 
2.S 

3.0 
3-5 
4.0 
4.S 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

982 

87  040 

099 

988 
046 
lOi 

994 
052 
111 

999 
058 
116 

*005 
064 
122 

*011 
070 

128 

*017 
075 
134 

*023 
081 
140 

*029 
087 
146 

*035 
093 
151 

44 
45 
46 

157 
216 
274 

163 
221 
280 

169 

227 
286 

175 
233 
291 

181 
239 
297 

186 
245 
303 

192 
251 
309 

198 
256 
315 

204 
262 
320 

210 

268 

•  326 

47 

48 

49 

750 

332 
390 
448 

338 
396 

454 

344 
402 
460 

349 
408 
466 

355 
413 
471 

361 
419 

477 

367 
425 
483 

373 
431 
489 

379 
437 
495 

384 
442 
500 

506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

14 


Logarithms  of  Numbers 

750-800 


If. 

L.  0 

1 

2 

3 

4 

5 

6 

7   8 

9 

Prop.  Pts. 

750 

51 
52 
53 

87  506 

512 

518 

523 

529 

535 

541 

547  1  552 

558 

564 
622 
679 

570 
628 
685 

576 
633 
691 

581 
639 
697 

587 
645 
703 

593 
651 
708 

599 
656 
714 

604 
662 
720 

610 

1  668 
726 

616 
674 
731 

54 
55 
56 

737 
795 

852 

743 
800 
858 

749 
806 
864 

754 
812 
869 

760 

818 
875 

766 
823 
881 

772 
829 
887 

777 
835 
892 

783 
841 
898 

789 
846 
904 

57 
58 
59 
760 
61 
62 
63 

910 

967 

88  024 

915 
973 
030 

921 
978 
036 

927 
984 
041 

933 
990 
047 

938 
996 
053 

944 

*001 

058 

950 

*007 

064 

955 

*013 

070 

%1 

*018 

076 

081 

087 

093 

098 

104 

110 

116 

121 

127 
184 
241 
298 

133 

138 
195 
252 

144 
201 

258 

150 
207 
264 

156 
213 
270 

161 
218 

275 

167 
224 
281 

173 
230 
287 

178 
235 
292 

190 
247 
304 

6 

64 
65 
66 

309 
366 
423 

315 
372 
429 

321 
377 
434 

326 
383 
440 

332 
389 
446 

338 
395 
451 

343 
400 
457 

349 
406 
463 

355 
412 
468 

360 
417 
474 

I 

2 

3 

4 
5 

0.6 

1.2 

1.8 
2.4 
3.0 
1  f> 

67 
68 
69 
770 
71 
72 
73 

480 
536 
593 

485 
542 
598 

491 

547 
604 

497 

553 
610 

502 
559 
615 

508 
564 
621 

513 

570 
627 

519 

576 
632 

525 
581 
638 

530 
587 
643 

7 
8 
9 

4.2 

4.8 
S-4 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

705 
762 
818 

711 
767 
824 

717 
773 
829 

722 
779 
83i 

728 
784 
840 

734 
790 
846 

739 
795 
852 

745 
801 

857 

750 
807 
863 

756 
812 
868 

74 
75 
76 

874 
930 
986 

880 
936 
992 

885 
941 
997 

891 

947 

*003 

897 

953 

*009 

902 

958 

*014 

908 

964 

*020 

913 

969 

*025 

919 

975 

*031 

925 

981 

*037 

77 
78 
79 

780 

81 
82 
83 

89  042 
098 
154 

048 
104 
159 

053 
109 
16i 

059 
115 
170 

064 
120 
176 

070 
126 
182 

076 
131 

187 

081 
137 
193 

087 
143 
198 

092 
148 
204 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 

265 
321 
376 

271 
326 
382 

276 
332 
387 

282 
337 
393 

287 
343 
398 

293 
348 
404 

298 
354 
409 

304 
360 
415 

310 
365 
421 

315 
371 
426 

I 

6 

o.S 

84 

85 
86 

432 
487 
542 

437 
492 
548 

443 
498 
553 

448 
504 
559 

454 
509 
564 

459 
515 
570 

465 
520 

575 

470 

526 
581 

476 
531 
586 

481 

537 
592 

3 

4 
S 
6 
7 

2.0 
2-S 

3.0 
3-5 

87 
88 
89 
790 
91 
92 
93 

597 
653 
708 

603 

658 
713 

609 
664 
719 

614 
669, 

724 

620 

675 
730 

625 
680 

735 

631 

686 
741 

636 
691 
746 

642 
697 
752 

647 
702 

757 

8 
9 

4.0 
4-5 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

818 
873 
927 

823 
878 
933 

829 
883 
938 

834 
889 
9H 

840 
894 
949 

845 
900 
95i 

851 
905 
960 

856 
911 
966 

862 
916 
971 

867 
922 
977 

94 
95 
96 

982 

90  037 

091 

988 
042 
097 

993 
048 
102 

998 
053 
108 

*004 
059 
113 

*009 
064 
119 

*015 
069 
124 

*020 
075 
129 

*026 
080 
135 

*031 
086 
140 

97 

98 

99 

800 

146 
200 

255 

151 
206 
260 

157 
211 
266 

162 
217 
271 

168 
222 
276 

173 
227 

282 

179 
233 

287 

184 
238 
293 

189 
244 
298 

195 
249 
304 

309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

15 


Logarithms  of  Numbers 

800-850 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

800 

01 
02 
03 

90  309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

363 
417 
472 

369 
423 

477 

374 
428 
482 

380 
434 

488 

385 
439 
493 

390 
445 
499 

396 
450 
504 

401 

455 
509 

407 
461 

515 

412 

466 
520 

04 
05 
06 

526 
580 
634 

531 

585 
639 

536 
590 
644 

542 
596 
650 

547 
601 
655 

553 
607 
660 

558 
612 
666 

563 
617 
671 

569 
623 
677 

574 
628 
682 

07 
OS 
09 
810 
11 
12 
13 

687 
741 
795 
849 

693 

747 
800 

698 

752 
806 

703 

757 
811 

709 
763 
816 

714 

768 
822 

720 
773 

827 

725 
779 
832 

730 

784 
838 

736 

789 
843 

854 

859 

865 

870 
924 
977 
030 

875 
929 
982 
036 

881 
934 
988 
041 

886 

891 

897 

950 

*004 

057 

902 

956 

91  009 

907 
961 
014 

913 
966 
020 

918 
972 
025 

940 
993 
046 

945 
998 
052 

0 

14 
15 
16 

062 
116 
169 

068 
121 
174 

073 
126 
180 

078 
132 
185 

084 
137 
190 

089 
142 
196 

094 
148 
201 

100 
153 
206 

105 
158 
212 

110 
164 

217 

2 

3 

4 

S 
5 

1.2 

1.8 

2.4 

3-0 

1  f, 

17 

18 

19 

820 

21 
22 
23 

222 
275 
328 

228 
281 
334 

233 
286 
339 

238 
291 
344 

243 
297 
350 

249 
302 
355 

254 
307 
360 
413 

259 
312 
365 

265 
318 
371 

270 
323 
376 
429 

7 
8 
9 

4.2 
4.8 
5-4 

381 

387 

392 

397 
450 
503 
556 

403 

408 

418 

424 

434 
487 
540 

440 
492 
545 

445 
498 
551 

455 
508 
561 

461 
514 
566 

466 
519 
572 

471 
524 

577 

477 
529 

582 

482 
535 
587 

24 
25 
26 

593 
645 
698 

598 
651 
703 

603 
656 
709 

609 
661 

714 

614 
666 
719 

619 

672 

724 

624 
677 
730 

630 
682 

735 

635 
687 
740 

640 
693 
745 

27 
28 
29 
830 
31 
32 
33 

751 
803 

855 

756 
808 
861 

761 
814 
866 

766 
819 
871 

772 
824 
876 

777 
829 
882 

782 
834 
887 

787 
840 
892 

793 
845 
897 
950 

798 
850 
903 

955 

908 

913 

918 

924 

929 

934 

939 

944 

960 

92  012 

06i 

965 
018 
070 

971 
023 
075 

976 
028 
080 

981 
033 
085 

986 
038 
091 

991 
044 
096 

997 
049 
101 

*002 
054 
106 

*007 
059 
111 

I 

5 

o.S 

34 
35 
36 

117 
169 

221 

122 
174 
226 

127 
179 
231 

132 
184 
236 

137 
189 
241 

143 
195 
247 

148 
200 
252 

153 
205 
257 

158 
210 
262 

163 
215 
267 

3 
4 
S 
6 

2.0 

2-S 

3-0 

37 
38 
39 
840 
41 
42 
43 

273 
324 
376 

278 
330 
381 

283 
335 

387 

288 
340 
392 

293 
345 
397 

298 
350 
402 

304 

355 
407 

309 
361 

412 

314 
366 
418 

319 
371 
423 

8 
9 

4.0 
4.S 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

480 
531 

583 

485 
536 
588 

490 
542 
593 

495 
547 
598 

500 
552 
603 

505 
557 
609 

511 
562 
614 

516 
567 
619 

521 
572 
624 

526 
578 
629 

44 
45 
46 

634 
686 

737 

639 
691 
742 

645 
696 

747 

650 
701 

752 

655 
706 

758 

660 
711 
763 

665 
716 
768 

670 
722 
773 

675 

727 
778 

681 
732 
783 

47 

48 

49 

850 

788 
840 
891 
942 

793 
845 
896 

799 
850 
901 

804 
855 
906 

809 
860 
911 

814 
865 
916 

819 
870 
921 

824 
875 
927 

829 
881 
932 

834 
886 
937 

947 

952 

957 

962 

967 

973 

978 

983 

988 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

16 


Logarithms  of  Numbers 

850-900 


N. 

L.  0 

1 

2 

3    4 

5 

6 

7 

8 

9 

Prop.  Pts. 

850 

51 

52 
53 

92  942 
993 

93  044 
095 

947  952 

957 

962 

967 

973 

978 

983 

988 

*039 

090 

141 

998 
(H9 
100 

*003 
054 
105 

*008; 
059 
110 

*013 
064 
115 

*018 
069 
120 

*024 
075 
125 

*029 
OSO 
131 

*034 
085 
136 

54 
55 
56 

146 
197 
247 

151 

202 
252 

156 
207 
258 

161 
212 
263 

166 

217 
268 

171 

222 
273 

176 

227 
278 

181 
232 
283 

186 

237 
288 

192 

242 
293 

57 
58 
59 
860 
61 
62 
63 

298 
349 
399 

303 

354 
404 

308 
359 
409 

313 
364 
414 

318 
369 
420 

323 
374 

425 

328 
379 
430 

334 

384 
435 

339 
389 
440 

344 
394 
445 

I 

2 

3 

4 
S 
6 
7 
8 

6 

0.6 

1.2 

1.8 

2.4 

3.0 
3.6 

4.2 

4.8 

450 

455 

460 

465 

470 

475  480 

485 

490 

495 

500 
551 
601 

505 
556 
606 

510 
561 
611 

515 
566 
616 

520 
571 
621 

526 
576 
626 

531 
581 
631 

536 
586 
636 

541 
591 
641 

546 
596 
646 

64 
65 
66 

651 
702 

752 

656 
707 
757 

661 
712 
762 

666 

717 
767 

671 
722 

772 

676 

727 

777 

682 
732 

782 

687 
737 

787 

692 

742 
792 

697 
747 
797 

9 

S-4 

67 

68 

69 

870 

71 
72 
73 

802 
852 
902 
952 

807 
857 
907 

812 
862 
912 

817 
867 
917 

822 
872 
922 

827 
877 
927 

832 
882 
932 

837 
887 
937 

842 
892 
942 

847 
897 
947 

957 

962 

967 

972 

977 

982 

987 

992 

997 

94  002 
052 
101 

007 
057 
106 

012 
062 
111 

017 
067 
116 

022 
072 
121 

027 
077 
126 

032 
082 
131 

037 
086 
136 

042 
091 
141 

047 
096 
146 

I 

5 

o.S 

74 
75 
76 

151 
201 
250 

156 
206 

255 

161 

211 
260 

166 
216 
265 

171 
221 
270 

176 
226 

275 

ISrl 

231 
280 

186 
236 

285 

191 

240 
290 

196 

245 
295 

2 

3 

4 
S 
6 

I.O 

i.S 

2.0 
2.S 

3-0 

77 

78 

79 

880 

81 
82 
83 

300 
349 
399 

305 
354 
404 

310 
359 
409 

315 
364 
414 

320 
369 
419 

325 

374 
424 

330 
379 

429 
478 

335 

384 
433 

340 

389 
438 

345 
394 
443 

7 
8 
9 

3-S 
4.0 
4-S 

448 

453 

458 

463 

468 

473 

483 

488 
537 
586 
635 

493 

498 
547 
596 

503 
552 
601 

507 
557 
606 

512 
562 
611 

517 
567 
616 

522 
571 
621 

527 
576 
626 

532 
581 
630 

542 
591 
640 

84 
85 
86 

645 
694 

743 

650 
699 

748 

655 
704 

753 

660 

709 

758 

665 
714 
763 

670 
719 

768 

675 

724 
773 

680 
729 
778 

685 
734 
783 

689 
738 

787 

4 

87 
88 
89 
890 
91 
92 
93 

792 
841 
890 

797 
846 
895 

802 
851 
900 

807 
856 
905 

812 
861 
910 

817 
866 
915 

822 
871 
919 

827 
876 
924 

832 
880 
929 

836 
885 
934 
983 

I 

2 

3 
4 
S 
6 
7 
8 
9 

0.4 
0.8 

1.2 

1.6 

2.0 

2.4 
2.8 

3-2 
3.6 

939 

944 

949 

954 

959 

*007 

056 

105 

963 

968 

973 

978 

988 

95  036 

085 

993 
041 
090 

998 
046 
095 

*002 
051 
100 

*012 
061 
109 

*017 
066 
114 

*022 
071 
119 

*027 
075 
124 

*032 
080 
129 

94 
95 
96 

134 
182 
231 

139 

187 
236 

143 
192 
240 

148 
197 

245 

153 
202 
250 

158 
207 
255 

163 
211 
260 

168 
216 
265 

173 
221 
270 

177 
226 
274 

97 

98 

99 

900 

279 
328 
376 

284 
332 
381 

289 
337 
386 

294 
342 
390 

299 
347 
395 

303 
352 
400 

308 

357 
405 

313 
361 
410 

318 
366 
415 
463 

323 
371 
419 

468 

424 

429 

1  434  i  439 

444 

448  !  453 

458 

N. 

L.  0 

1 

2    3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

17 


Logarithms  of  Numbers 

900-950 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

900 

01 
02 
03 

95  424 
472 
521 
569 

429 
477 

525 
574 

434 
482 
530 
578 

439 

444 

448 

453 

458 

463 
511 
559 
607 

468 

Tl6 

564 

612 

487 
535 
583 

492 
540 
588 

497 
545 
593 

501 
550 
598 

506 
554 
602 

04 
05 
06 

617 
665 
713 

622 
670 
718 

626 
674 

722 

631 
679 

727 

636 

684 
732 

641 
689 

737 

646 
694 
742 

650 
698 
746 

655 
703 
751 

660 
708 
756 

07 
08 
09 
910 
11 
12 
13 

761 
809 
856 

766 
813 
861 

770 

818 
866 

775 
823 
871 

780 
828 
875 

785. 
832 
880 

789 

837 
885 

794 

842 
890 

799 

847 
895 

804 
852 
899 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

952 

999 

96  047 

957 

*004 

052 

961 

*009 

057 

966 

*014 

061 

971 

*019 

066 

976 

*023 

071 

980 

*028 

076 

985 

*033 

080 

990 

*038 

085 

995 

*042 

090 

5 

14 
15 
16 

095 
142 
190 

099 
147 
194 

104 
152 
199 

109 
156 
204 

114 
161 
209 

118 
166 
213 

123 
171 
218 

128 
175 
223 

133 
180 
227 

137 
185 
232 

I 

2 

3 

4 

5 
5 

o.S 

I.O 

i-S 

2.0 

2.5 

17 

18 

19 

920 

21 
22 
23 

237 
284 
332 

242 
289 
336 

246 
294 
341 

251 
298 
346 

256 
303 
350 

261 
308 
355 

265 
313 
360 

270 
317 
365 

275 
322 
369 

280 
327 
374 

7 
8 
9 

3.S 
4.0 
4-5 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

426 
473 
520 

431 

478 
525 

435 
483 
530 

440 
487 
534 

445 
492 
539 

450 
497 
544 

454 
501 
548 

459 
506 
553 

464 
511 
558 

468 
515 
562 

24 
25 
26 

567 
614 
661 

572 
619 
666 

577 
624 
670 

581 
628 
675 

586 
633 
680 

591 

638 
685 

595 
642 
689 

600 
647 
694 

605 
652 
699 

609 
656 
703 

27 
28 
29 
930 
31 
32 
33 

708 

755 
802 

713 

759 
806 

717 
764 
811 

722 
769 
816 

727 
774 
820 

731 

778 
825 

736 
783 
830 

741 
788 
834 

745 
792 
839 

750 
797 

844 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

895 
942 
988 

900 
946 
993 

904 
951 
997 

909 

956 

*002 

914 

960 

*007 

918 

965 

*011 

923 

970 

*016 

928 

974 

*021 

932 

979 
*025 

937 

984 

*030 

I 

4 

0.4 
0.8 

1.2 

1.6 

2.0 
2.4 
2.8 

34 
35 
36 

97  035 
081 
128 

039 
086 
132 

044 
090 
137 

049 
095 
142 

053 
100 
146 

058 
104 
151 

063 
109 

155 

067 
114 
160 

072 
118 
165 

077 
123 
169 

3 
4 
5 
6 

37 
38 
39 
940 
41 
42 
43 

174 
220 
267 

179 
225 
271 

183 
230 
276 

188 
234 
280 

192 
239 

285 

197 
243 
290 

£02 
248 
294 

206 
253 
299 

211 

257 
304 

216 
262 
308 

8 
0 

3.6 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 

359 
405 
451 

364 
410 
456 

368 
414 
460 

373 
419 
465 

377 
424 
470 

382 
428 
474 

387 
433 
479 

391 

437 
483 

396 

442 
488 

400 
447 
493 

44 
45 
46 

497 
543 
589 

502 

548 
594 

506 

552 
598 

511 

557 
603 

5K) 
562 
607 

520 
566 
612 

525 
571 
617 

529 

575 
621 

534 
580 
626 

539 

585 
630 

47 

48 

49 

950 

635 
681 

727 

640 
685 
731 

644 
690 
736 

649 
695 
740 

653 
699 

745 

658 
704 

749 

663 

708 
754 

667 
713 
759 

672 
717 
763 

676 

722 
768 

772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

18 


LOGAKITHMS    OF    NuMBERS 

950-1000 


N. 

L.  0 

1 

2  I  3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

950 

51 
52 
53 

97  772 

777 

782 

786 

791 
836 
882 
928 

795 
841 
886 
932 

800 
845 
891 
937 

804 
850 
896 
941 

809 
855 
900 
946 

813 
859 
905 
950 

818 
864 
909 

823 
868 
914 

827 
873 
918 

1  832 
877 
923 

54 
55 
56 

955 

98  000 

046 

959 
005 
050 

964 

009 
055 

968 
014 
059 

973 
019 
064 

978 
023 
068 

982 
028 
073 

987 
032 

078 

991 
037 
082 

9% 
041 
087 

• 

57 
58 
59 
960 
61 
62 
63 

091 
137 

182 

096 
141 
186 

100 
146 
191 

105 
150 
195 

109 

155 
200 

114 

159 
204 

118 
164 
209 
254 

123 

168 
214 

127 
173 
218 

132 

177 
223 
268 
313 
358 
403 

227 

232 

236 

241 

245 

250 

259 

263 
308 
354 
399 

272 
318 
363 

277 
322 
367 

281 
327 
372 

286 
331 
376 

290 
336 
381 

295 
340 
385 

299 
345 
390 

304 
349 
394 

6 

64 
65 
66 

408 
453 
498 

412 
457 
502 

417 
462 
507 

421 
466 
511 

426 
471 
516 

430 
475 
520 

435 
480 
52i 

439 
484 
529 

444 
489 
534 

448 
493 
538 

I 

2 

3 

t 

6 
7 
8 
9 

o.S 

l.O 

i-S 

2,0 
2-5 

67 

68 

69 

970 

71 
72 
73 

543 
588 
632 

547 
592 
637 

552 
597 
641 

556 
601 
646 

561 
605 
650 

565 
610 
655 
700 

570 

614. 

659 

574 
619 
664 
709 
753 
798 
843 

579 
623 
668 
713 

758 
802 
847 

583 
628 
673 
717 
762 
807 
851 

3-5 
4.0 
4-5 

677 

682 

686  1  691 

695 

70  V 

722 
767 
811 

726 
771 
816 

731 
776 
820 

735 

780 
825 

740 
784 
829 

744 
789 
834 

749 
793 
838 

74 
75 
76 

856 
900 
945 

860 
905 
949 

865 
909 
954 

869 
914 
958 

874 
918 
963 

878 
923 
967 

883 
927 
972 

887 
932 
976 

892 
936 
981 

8% 
941 
985 

77 

78 

79 

980 

81 
82 
83 

989 

99  034 

078 

123 

994 
038 
083 

998 
043 
087 

*003 
047 
092 

*007 
052 
0% 

*012 
056 
100 

*016 
061 
105 

*021 
065 
109 
154 

*025 
069 
114 

*029 
074 
118 

127 
171 
216 
260 

131  t  136 

140 
185 
229 
273 

145 

149 
193 
238 
282 

158 
202 
247 
291 

162 

167 
211 
255 

176 
220 
264 

180 
224 
269 

189 
233 
277 

198 
242 
286 

207 
251 
295 

I 

4 

0.4 

84 
85 
86 

300 
344 
388 

304 
348 
392 

308 
352 
396 

313 
357 
401 

317 
361 
405 

322 
366 
410 

326 
370 
414 

330 
374 
419 

335 
379 
423 

339 
383 
427 

2 

3 
4 

1 

0.8 

1.2 

1.6 

2.0 
2-4 

87 
88 
89 
990 
9] 
92 
93 

432 
476 

520 

436 

480 
524 

441 

484 

528 

445 
489 
533 

449 
493 
537 

454 
498 
542 

458 
502 
546 
590 
634 
677 
721 

463 
506 
550 
594 
638 
682 
726 

467 
511 

555 

471 
515 
559 

8 
9 

3.2 
3.6 

564 

568 

572 

577 

581 

585 

599 

603 

607 
651 
69i 

612 
656 
699 

616 
660 
704 

621 
66V 
708 

6^5 
669 
712 

629 
673 
717 

642 
686 
730 

647 
691 
734 

94 
95 
96 

739 

782 
826 

743 

787 
830 

747 
791 

835 

752 
795 
839 

756 
800 

843 

760 
804 

848 

765 
808 
852 

769 
813 
856 

774 
817 
861 

778 
822 
865 

97 

98 

99 

1000 

870 
913 
957 

874 
917 
%1 

878 
922 
965 

883  887 
926  930 
970  974 

891 
935 
978 

896 
939 
983 

900 
944 
987 

9M 
948 
991 

909 
952 
996 

00  000 

004 

009 

013 

017 

022 

026 

030 

035 

039 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

19 


TABLE  II 


MYE -PLACE  LOGARITHMS 


OF  THE 


TRIGONOMETRIC  FUIVCTIONS 


91 


Table  II  contains  the  common  logarithms  of  the  trigonometric 
functions  for  angles  from  0°  to  90°  as  follows : 

(1)  log  sin  a  =  log  tan  a,  for  every  second  0°  to  3',  and  log  cos  /3  = 
log  cot  /3,  for  every  second,  89°  57'  to  90°,  page  22. 

(2)  logarithms  of  sine,  tangent,  and  cosine  for  every  10"  from  0° 
to  2° ;  also  the  logarithms  of  sine,  cosine,  and  cotangent  for  every 
10"  from  88°  to  90°,  pages  23-28. 

(3)  logarithms  of  each  function  for  every  1'  from  0°  to  90°,  pages 
29-73. 


log  sin  =  log  t 

an 

0° 

// 

0' 

1' 

2' 

II 

II 

0' 

1' 

2' 

II 

0 

—  00 

6.46  373 

6.76  476 

60 

30 

6.16  270 

6.63  982 

6.86  167 

30 

1 

4.68  557 

6.47  090 

6.76  836 

59 

31 

6.17  694 

6.64  462 

6.86  455 

29 

2 

4.98  660 

6.47  797 

6.77  193 

58 

32 

6.19  072 

6.64  936 

6.86  742 

28 

3 

5.16  270 

6.48  492 

6.77  548 

57 

33 

6.20  409 

6.65  406 

6.87  027 

27 

4 

5.28  763 

6.49  175 

6.77  900 

56 

34 

6.21  705 

6.65  870 

6.87  310 

26 

5 

5.38  454 

6.49  849 

6.78  248 

55 

35 

6.22  964 

6.66  330 

6.87  591 

25 

6 

5.46  373 

6.50  512 

6.78  595 

54 

36 

6.24  188 

6.66  785 

6.87  870 

24 

7 

5.53  067 

6.51  165 

6.78  938 

53 

37 

6.25  378 

6.67  235 

6.88  147 

23 

8 

5.58  866 

6.51  808 

6.79  278 

52 

38 

6.26  536 

6  67  680 

6.88  423 

22 

9 

5.63  982 

6.52  442 

6.79  616 

51 

39 

6.27  664 

6.68  121 

6.88  697 

21 

10 

5.68  557 

6.53  067 

6.79  952 

50 

40 

6.28  763 

6.68  557 

6.88  969 

20 

11 

5.72  697 

6.53  683 

6.80  285 

49 

41 

6.29  836 

6.68  990 

6.89  240 

19 

12 

5.76  476 

6.54  291 

6.80  615 

48 

42 

6.30  882 

6.69  418 

6.89  509 

18 

13 

5.79  952 

6.54  890 

6.80  943 

47 

43 

6.31  904 

6.69  841 

6.89  776 

17 

14 

5.83  170 

6.55  481 

6.81  268 

46 

44 

6.32  903 

6.70  261 

6.90  042 

16 

15 

5.86  167 

6.56  064 

6.81  591 

45 

45 

6.33  879 

6.70  676 

6.90  306 

15 

16 

5.88  969 

6.56  639 

6.81  911 

44 

46 

6.34  833 

6.71  088 

6.90  568 

14 

17 

5.91  602 

6.57  207 

6.82  230 

43 

47 

6.35  767 

6.71  496 

6.90  829 

13 

18 

5.94  085 

6.57  767 

6.82  545 

42 

48 

6.36  682 

6.71  900 

6.91  088 

12 

19 

5.96  433 

6.58  320 

6.82  859 

41 

49 

6.37  577 

6.72  300 

6.91  346 

11 

20 

5.98  660 

6.58  866 

6.83  170 

40 

50 

6.38  454 

6.72  697 

6.91  602 

10 

21 

6.00  779 

6.59  406 

6.83  479 

39 

51 

6.39  315 

6.73  090 

6.91  857 

9 

22 

6.02  800 

6.59  939 

6.83  786 

38 

52 

6.40  158 

6.73  479 

6.92  110 

8 

23 

6.04  730 

6.60  465 

6.84  091 

37 

53 

6.40  985 

6.73  865 

6.92  362 

7 

24 

6.06  579 

6.60  985 

6.84  394 

36 

54 

6.41  797 

6.74  248 

6.92  612 

6 

25 

6.08  351 

6.61  499 

6.84  694 

35 

55 

6.42  594 

6.74  627 

6.92  861 

5 

26 

6.10  055 

6.62  007 

6.84  993 

34 

56 

6.43  376 

6.75  003 

6.93  109 

4 

27 

6.11  694 

6.62  509 

6.85  289 

33 

57 

6.44  145 

6.75  376 

6.93  355 

3 

28 

6.13  273 

6.63  006 

6.85  584 

32 

58 

6.44  900 

6.75  746 

6.93  599 

2 

29 

6.14  797 

6.63  496 

6.85  876 

31 

59 

6.45  643 

6.76  112 

6.93  843 

1 

30 

6.16  270 

6.63  982 

6.86  167 

30 

60 

6.46  373 

6.76  476 

6.94  085 

0 

// 

59' 

58' 

57' 

II 

II 

59' 

58' 

57' 

II 

1 

81 

)° 

log  COS  =  log  cot 

J 

22 


Logarithms  of  the  Trigonometric  Functions 


0 

>° 

1 

/  // 

log  sill   log  tan 

log  cos 

//  / 

/  // 

log  sin 

log  tan 

log  cos 

//  / 

0  0 

—  00     —  00 

10.00000 

060 

10  0 

7.46  373 

7.46  373 

10.00000 

0  50 

10 

5.68  557  5.68  557 

10.00000 

50 

10 

7.47  090 

7.47  091 

10.00000 

50 

20 

5.98  660  i  5.98  660 

10.00000 

40 

20 

7.47  797 

7.47  797 

10.00000 

40 

30 

6.16  270  6.16  270 

10.00000 

30 

30 

7.48  491 

7.48  492 

10.00000 

30 

40 

6.28  763  i  6.28  763 

10.00000 

20 

40 

7.49  175 

7.49  176 

10.00000 

20 

50 

6.38  454  1  6  38  454 

10.00000 

10 

50 

7.49  849 

7.49  849 

10.00000 

10 

1  0 

6.46  373  6.46  373 

10.JOOOOO 

059 

110 

7.50  512 

7.50  512 

10.00000 

049 

10 

6.53  067 

6.53  067 

10.00000 

50 

10 

7.51  165 

7.51  165 

10.00000 

50 

20 

6.58  866 

6.58  866 

10.00000 

40 

20 

7.51  808 

7.51  809 

10.00000 

40 

30 

6.63  982 

6.63  982 

10.00000 

30 

30 

7.52  442 

7.52  443 

10.00000 

30 

40 

6.68  557 

6.68  557 

10.00000 

20 

40 

7.53  067 

7.53  067 

10.00000 

20 

50 

6.72  697 

6.72  697 

10.00000 

10 

50 

7.53  683 

7.53  683 

10.00000 

10 

2  0 

6.76  476 

6.76  476 

10.00000 

058 

12  0 

7.54  291 

7.54  291 

10.00000 

048 

10 

6.79  952 

6.79  952 

10.00000 

50 

10 

7.54  890 

7.54  890 

10.00000 

50 

20 

6.83  170 

6.83  170 

10.00000 

40 

20 

7.55  481 

7.55  481 

10.00000 

40 

30 

6.86  167 

6.86  167 

10.00000 

30 

30 

7.56  064 

7.56  064 

10.00000 

30 

40 

6.88  969 

6.88  969 

10.00000 

20 

40 

7.56  639 

7.56  639 

10.00000 

20 

50 

6.91  602 

6.91  602 

10.00000 

10 

50 

7.57  206 

7.57  207 

10.00000 

10 

3  0 

6.94  085 

6.94  085 

10.00000 

057 

13  0 

7.57  767 

7.57  767 

10.00000 

0  47 

10 

6.96  433 

6.96  433 

10.00000 

50 

10 

7.58  320 

7.58  320 

10.00000 

50 

20 

6.98  660 

6.98  661 

10.00000 

40 

20 

7.58  866 

7.58  867 

10.00000 

40 

30 

7.00  779 

7.00  779 

10.00000 

30 

30 

7.59  406 

7.59  406 

10.00000 

30 

40 

7.02  800 

7.02  800 

10.00000 

20 

40 

7.59  939 

7.59  939 

10.00000 

20 

50 

7.04  730 

7.04  730 

10.00000 

10 

50 

7.60  465 

7.60  466 

10.00000 

10 

4  0 

7.06  579 

7.06  579 

10.00000 

056 

140 

7.60  985 

7.60  986 

10.00000 

046 

10 

7.08  351 

7.08  352 

10.00000 

50 

10 

7.61  499 

7.61  500 

10.00000 

50 

20 

7.10  055 

7.10  055 

10.00000 

40 

20 

7.62  007 

7.62  008 

10.00000 

40 

30 

7.11  694 

7.11  694 

10.00000 

30 

30 

7.62  509 

7.62  510 

10.00000 

30 

40 

7.13  273 

7.13  273 

10.00000 

20 

40 

7.63  006 

7.63  006 

10.00000 

20 

50  7.14  797 

7.14  797 

10.00000 

10 

50 

7.63  496 

7.63  497 

10.00000 

10 

5  0  7.16  270 

7.16  270 

10.00000 

055 

15  0 

7.63  982 

7.63  982 

10.00000 

0  45 

10  7.17  694 

7.17  694 

10.00000 

50 

10 

7.64  461 

7.64  462 

10.00000 

50 

20  1  7.19  072 

7.19  073 

10.00000 

40 

20 

7.64  936 

7.64  937 

10.00000 

40 

30  7.20  409 

7.20  409 

10.00000 

30 

30 

7.65  406 

7.65  406 

10.00000 

30 

40  7.21  705 

7.21  705 

10.00000 

20 

40 

7.65  870 

7.65  871 

10.00000 

20 

50 

7.22  964 

7.22  964 

10.00000 

10 

50 

7.66  330 

7.66  330 

10.00000 

10 

6  0 

7.24  188 

7.24  188 

10.00000 

054 

160 

7.66  784 

7.66  785 

10.00000 

044 

10 

7.25  378 

7.25  378 

10.00000 

50 

10 

7.67  235 

7.67  235 

10.00000 

50 

20 

7.26  536 

7.26  536 

10.00000 

40 

20 

7.67  680 

7.67  680 

10.00000 

40 

30 

7.27  664 

7.27  664 

10.00000 

30 

30 

7.68  121 

7.68  121 

10.00000 

30 

40  7.28  763 

7.28  764 

10.00000 

20 

40 

7.68  557 

7.68  558 

9.99999 

20 

50 

7.29  836 

7.29  836 

10.00000 

10 

50 

7.68  989 

7.68  990 

9.99999 

10 

7  0 

7.30  882 

7.30  882 

10.00000 

053 

170 

7.69  417 

7.69  418 

9.99999 

043 

10 

7.31  904 

7.31  904 

10.00000 

50 

10 

7.69  841 

7.69  842 

9.99999 

50 

20 

7.32  903 

7.32  903 

10.00000 

40 

20 

7.70  261 

7.70  261 

9.99999 

40 

30 

7.33  879 

7.33  879 

10.00000 

30 

30 

7.70  676 

7.70  677 

9.99999 

30 

40 

7.34  833 

7.34  833 

10.00000 

20 

40 

7.71  088 

7.71  088 

9.99999 

20 

50 

7.35  767 

7.35  767 

10.00000 

10 

50 

7.71  496 

7.71  496 

9.99999 

10 

8  0 

7.36  682 

7.36  682 

10.00000 

052 

180 

7.71  900 

7.71  900 

9.99999 

042 

10 

7.37  577 

7.37  577 

10.00000 

50 

10 

7.72  300 

7.72  301 

9.99999 

50 

20 

7.38  454 

7.38  455 

10.00000 

40 

20 

7.72  697 

7.72  697 

9.99999 

40 

30 

7.39  314 

7.39  315 

10.00000 

30 

30 

7.73  090 

7.73  090 

9.99999 

30 

40 

7.40  158 

7.40  158 

10.00000 

20 

40 

7.73  479 

7.73  480 

9.99999 

20 

50 

7.40  985 

7.40  985 

10.00000 

10 

50 

7.73  865 

7.73  866 

9.99999 

10 

9  0 

7.41  797 

7.41  797 

10.00000 

0  51 

190 

7.74  248 

7.74  248 

9.99999 

0  41 

10 

7.42  594 

7.42  594 

10.00000 

50 

10 

7.74  627 

7.74  628 

9.99999 

50 

20 

7.43  376 

7.43  376 

10.00000 

40 

20 

7.75  003 

7.75  004 

9.99999 

40 

30 

7.44  145 

7.44  145 

10.00000 

30 

30 

7.75  376 

7.75  377 

9.99999 

30 

40 

7.44  900 

7.44  900 

10.00000 

20 

40 

7.75  745 

7.75  746 

9.99999 

20 

50 

7.45  643 

7.45  643 

10.00000 

10 

50 

7.76  112 

7.76  113 

9.99999 

10 

10  0 

7.46  373 

7.46  373 

10.00000 

0  50 

20  0 

7.76  475 

7.76  476 

9.99999 

040 

/  // 

log  cos 

log  cot 

log  sin 

//  / 

/  It 

log  COS 

log  cot 

log  sin 

//  / 

8 

9° 

1 

23 


Logarithms  of  the  Trigonometric  Functions 


0 

1 

/  // 

log  sin 

log  tan 

log  COS 

//  / 

/  // 

log  sill 

log  tan 

log  cos 

//  / 

20  0 

7.76  475 

7.76  476 

9.99  999 

0  40 

30  0 

7.94  084 

7.94 

086 

9.99  998 

0  30 

10 

7.76  836 

7.76  837 

9.99  999 

50 

10 

7.94  325 

7.94 

326 

9.99  998 

50 

20 

7.77  193 

7.77  194 

9.99  999 

40 

20 

7.94  564 

7.94 

566 

9.99  998 

40 

30 

7.77  548 

7.77  549 

9.99  999 

30 

30 

7.94  802 

7.94 

804 

9.99  998 

30 

40 

7.77  899 

7.77  900 

9.99  999 

20 

40 

7.95  039 

7.95 

040 

9.99  998 

20 

50 

7.78  248 

7.78  249 

9.99  999 

10 

50 

7.95  274 

7.95 

276 

9.99  998 

10 
0  29 

210 

7.78  594 

7.78  595 

9.99  999 

0  39 

310 

7.95  508 

7.95 

510 

9.99  998 

10 

7.78  938 

7.78  938 

9.99  999 

50 

10 

7.95  741 

7.95 

743 

9.99  998 

50 

20 

7.79  278 

7.79  279 

9.99  999 

40 

20 

7.95  973 

7.95 

974 

9.99  998 

40 

30 

7.79  616 

7.79  617 

9.99  999 

30 

30 

7.96  203 

7.96 

205 

9.99  998 

30 

40 

7.79  952 

7.79  952 

9.99  999 

20 

40 

7.96  432 

7.96 

434 

9.99  998 

20 

50 

7.80  284 

7.80  285 

9.99  999 

10 

50 

7.96  660 

7.96 

662 

9.99  998 

10 

22  0 

7.80  615 

7.80  615 

9.99  999 

0  38 

32  0 

7.96  887 

7.96 

889 

9.99  998 

0  28 

10 

7.80  942 

7.80  943 

9.99  999 

50 

10 

7.97  113 

7.97 

114 

9.99  998 

50 

20 

7.81  268 

7.81  269 

9.99  999 

40 

20 

7.97  337 

7.97 

339 

9.99  998 

40 

30 

7.81  591 

7.81  591 

9.99  999 

30 

30 

7.97  560 

7.97 

562 

9.99  998 

30 

40 

7.81  911 

7.81  912 

9.99  999 

20 

40 

7.97  782 

7.97 

784 

9.99  998 

20 

50 

7.82  229 

7.82  230 

9.99  999 

10 

50 

7.98  003 

7.98 

005 

9.99  998 

10 

23  0 

7.82  545 

7.82  546 

9.99  999 

0  37 

33  0 

7.98  223 

7.98 

225 

9.99  998 

0  27 

10 

7.82  859 

7.82  860 

9.99  999 

50 

10 

7.98  442 

7.98 

444 

9.99  998 

50 

20 

7.83  170 

7.83  171 

9.99  999 

40 

20 

7.98  660 

7.98 

662 

9.99  998 

40 

30 

7.83  479 

7.83  480 

9.99  999 

30 

30 

7.98  876 

7.98 

878 

9.99  998 

30 

40 

7.83  786 

7.83  787 

9.99  999 

20 

40 

7.99  092 

7.99 

094 

9.99  998 

20 

50 

7.84  091 

7.84  092 

9.99  999 

10 

50 

7.99  306 

7.99 

308 

9.99  998 

10 

24  0 

7.84  393 

7.84  394 

9.99  999 

0  36 

34  0 

7.99  520 

7.99 

522 

9.99  998 

0  26 

10 

7.84  694 

7.84  695 

9.99  999 

50 

10 

7.99  732 

7.99 

734 

9.99  998 

50 

20 

7.84  992 

7.84  993 

9.99  999 

40 

20 

7.99  943 

7.99 

946 

9.99  998 

40 

30 

7.85  289 

7.85  290 

9.99  999 

30 

30 

8.00  154 

8.00 

156 

9.99  998 

30 

40 

7.85  583 

7.85  584 

9.99  999 

20 

40 

8.00  363 

8.00 

365 

9.99  998 

20 

50 

7.85  876 

7. 85  877 

9.99  999 

10 

50 

8.00  571 

8.00 

574 

9.99  998 

10 

25  0 

7.86  166 

7.86  167 

9.99  999 

0  35 

35  0 

8.00  779 

8.00 

781 

9.99  998 

0  25 

10 

7.86  455 

7.86  456 

9.99  999 

50 

10 

8.00  985 

8.00 

987 

9.99  998 

50 

20 

7.86  741 

7.86  743 

9.99  999 

40 

20 

8.01  190 

8.01 

193 

9.99  998 

40 

30 

7.87  026 

7.87  027 

9.99  999 

30 

30 

8.01  395 

8.01 

397 

9.99  998 

30 

40 

7.87  309 

7.87  310 

9.99  999 

20 

40 

8.01  598 

8.01 

600 

9.99  998 

20 

50 

7.87  590 

7.87  591 

9.99  999 

10 

50 

8.01  801 

8.01 

803 

9.99  998 

10 

260 

7.87  870 

7.87  871 

9.99  999 

034 

360 

8.02  002 

8.02 

004 

9.99  998 

0  24 

10 

7.88  147 

7.88  148 

9.99  999 

50 

10 

8.02  203 

8.02 

205 

9.99  998 

50 

20 

7.88  423 

7.88  424 

9.99  999 

40 

20 

8.02  402 

8.02 

405 

9.99  998 

40 

30 

7.88  697 

7.88  698 

9.99  999 

30 

30 

8.02  601 

8.02 

604 

9.99  998 

30 

40 

7.88  969 

7.88  970 

9.99  999 

20 

40 

8,02  799 

8.02 

801 

9.99  998 

20 

50 

7.89  240 

7.89  241 

9.99  999 

10 

50 

8.02  996 

8.02 

998 

9.99  998 

10 

270 

7.89  509 

7.89  510 

9.99  999 

033 

37  0 

8.03  192 

8.03 

194 

9.99  997 

0  23 

10 

7.89  776 

7.89  777 

9.99  999 

50 

10 

8.03  387 

8.03 

390 

9.99  997 

50 

20 

7.90  041 

7.90  043 

9.99  999 

40 

20 

8.03  581 

8.03 

584 

9.99  997 

40 

30 

7.90  305 

7.90  307 

9.99  999 

30 

30 

8.03  775 

8.03 

777 

9-99  997 

30 

40 

7.90  568 

7.90  569 

9.99  999 

20 

40 

8.03  967 

8.03 

970 

9.99  997 

20 

50 

7.90  829 

7.90  830 

9.99  999 

10 

50 

8.04  159 

8.04 

162 

9.99  997 

10 

28  0 

7.91  088 

7.91  089 

9.99  999 

0  32 

38  0 

8.04  350 

8.04 

353 

9.99  997 

0  22 

10 

7.91  346 

7.91  347 

9.99  999 

50 

10 

8.04  540 

8.04 

543 

9.99  997 

50 

20 

7.91  602 

7.91  603 

9.99  999 

40 

20 

8.04  729 

8.04 

732 

9.99  997 

40 

30 

7.91  857 

7.91  858 

9.99  999 

30 

30 

8.04  918 

8.04 

921 

9.99  997 

30 

40 

7.92  110 

7.92  111 

9.99  998 

20 

40 

8.05  105 

8.05 

108 

9.99  997 

20 

50 

7.92  362 

7.92  363 

9.99  998 

10 

50 

8.05  292 

8.05 

295 

9.99  997 

10 

29  0 

7.92  612 

7.92  613 

9.99  998 

031 

39  0 

8.05  478 

8.05 

481 

9.99  997 

021 

10 

7.92  861 

7.92  862 

9.99  998 

50 

10 

8.05  663 

8.05 

666 

9.99  997 

50 

20 

7.93  108 

7.93  110 

9.99  998 

40 

20 

8.05  848 

8.05 

851 

9.99  997 

40 

30 

7.93  354 

7.93  356 

9.99  998 

30 

30 

8.06  031 

8  06 

034 

9.99  997 

30 

40 

7.93  599 

7.93  601 

9.99  998 

20 

40 

8.06  214 

8.06 

217 

9.99  997 

20 

50 

7.93  842 

7.93  844 

9.99  998 

10 

50 

8.06  396 

8.06 

399 

9.99  997 

10 
0  20 

30  0 

7.94  084 

7.94  086 

9.99  998 

030 

40  0 

8.06  578 

8.06 

581 

9.99  997 

/  // 

log  COS 

log  cot 

log  sin 

ti  f 

/  // 

log  COS 

log  cot 

log  sin 

//  / 

81 

)° 

1 

24 


Logarithms  of  the  Tkigoxometric  Functions 


0 

1 

'  "  log  sin  1  log  tan   log  cos   "  ' 

/  // 

log  sin 

log  tan 

log  cos 

//  / 

40  0 

8.06  578 

8.06  581 

9.99  997 

0  20 

50  0 

8.16  268 

8.16  273 

9.99  995 

010 

10 

8.06  758 

8.06  761 

9.99  997 

50 

10 

8.16  413 

8.16  417 

9.99  995 

50 

20 

8.06  938 

8.06  941 

9.99  997 

40 

20 

8.16  557 

8.16  561 

9.99  995 

40 

30 

8.07  117 

8.07  120 

9.99  997 

30 

30 

8.16  700 

8.16  705 

9.99  995 

30 

40 

8.07  295 

8.07  299 

9.99  997 

20 

40 

8.16  843 

8.16  848 

9.99  995 

20 

50 

8.07  473 

8.07  476 

9.99  997 

10 

50 

8.16  986 

8.16  991 

9.99  995 

10 

410  1  8.07  650 

8.07  653 

9.99  997 

019 

510 

8.17  128 

8.17  133 

9.99  995 

0  9 

10 

8.07  826  8.07  829 

9.99  997 

50 

10 

8.17  270 

8.17  275 

9.99  995 

50 

20 

8.08  002 

8.08  005 

9.99  997 

40 

20 

8.17  411 

8.17  416 

9.99  995 

40 

30 

8.08  176 

8.08  180 

9.99  997 

30 

30 

8.17  552 

8.17  557 

9.99  995 

30 

40 

8.08  350 

8.08  354 

9.99  997 

20 

40 

8.17  692 

8.17  697 

9.99  995 

20 

50 

8.08  524 

8.08  527 

9.99  997 

10 

50 

8.17  832 

8.17  837 

9.99  995 

10 

42  0 

8.08  696 

8.08  700  9.99  997 

018 

52  0 

8.17  971 

8.17  976 

9.99  995 

0  8 

10 

8.08  868 

8.08  872  9.99  997 

50 

10 

8.18  110 

8.18  115 

9.99  995 

50 

20 

8.09  040 

8.09  043  9.99  997 

40 

20 

8.18  249 

8.18  254 

9.99  995 

40 

30 

8.09  210 

8.09  214 

9.99  997 

30 

30 

8.18  387 

8.18  392 

9.99  995 

30 

40 

8.09  380 

8.09  384 

9.99  997 

20 

40 

8.18  524 

8.18  530 

9.99  995 

20 

50 
43  0 

8.09  550 

8.09  553 

9-99  997 

10 

50 

8.18  662 

8.18  667 

9.99  995 

10 

8.09  718 

8.09  722 

9.99  997 

017 

53  0 

8.18  798 

8.18  804 

9.99  995 

0  7 

10  8.09  886 

8.09  890 

9.99  997 

50 

10 

8.18  935 

8.18  940 

9.99  995 

50 

20 

8.10  054 

8.10  057 

9.99  997 

40 

20 

8.19  071 

8.19  076 

9.99  995 

40 

30 

8.10  220 

8.10  224 

9.99  997 

30 

30 

8.19  206 

8.19  212 

9.99  995 

30 

40 

8 10  386 

8.10  390 

9.99  997 

20 

40 

8.19  341 

8.19  347 

9.99  995 

20 

50 
44  0 

8.10  552 

8.10  555 

9.99  996 

10 

50 

8.19  476 

8.19  481 

9.99  995 

10 

8.10  717 

8.10  720 

9.99  996 

016 

54  0 

8.19  610 

8.19  616 

9.99  995 

0  6 

10 

8.10  881 

8.10  884 

9.99  996 

50 

10 

8.19  744 

8.19  749 

9.99  995 

50 

20  8.11  044 

8.11  048 

9.99  996 

40 

20 

8.19  877 

8.19  883 

9.99  995 

40 

30  8.11  207 

8.11  211 

9.99  996 

30 

30 

8.20  010 

8.20  016 

9.99  995 

30 

40  8.11  370 

8.11  373 

9.99  996 

20 

40 

8.20  143 

8.20  149 

9.99  995 

20 

50  8.11  531 

8.11  535 

9.99  996 

10 

50 

8.20  275 

8.20  281 

9.99  994 

10 

45  0  i  8.11  693 

8.11  696 

9.99  996 

015 

55  0 

8.20  407 

8.20  413 

9.99  994 

0  5 

10 

8.11  853 

8.11  857 

9.99  996 

50 

10 

8.20  538 

8.20  544 

9.99  994 

50 

20 

8.12  013 

8.12  017 

9.99  996 

40 

20 

8.20  669 

8.20  675 

9.99  994 

40 

30 

8.12  172 

8.12  176 

9.99  996 

30 

30 

8.20  800 

8.20  806 

9.99  994 

30 

40  8.12  331 

8.12  335 

9.99  996 

20 

40 

8.20  930 

8.20  936 

9.99  994 

20 

50 

8.12  489 

8.12  493 

9.99  996 

10 

50 

8.21  060 

8.21  066 

9.99  994 

10 

46  0 

8.12  647 

8.12  651 

9.99  996 

014 

56  0 

8.21  189 

8.21  195 

9.99  994 

0  4 

10 

8.12  804 

8.12  808 

9.99  996 

50 

10 

8.21  319 

8.21  324 

9.99  994 

50 

20 

8.12  961 

8.12  965 

9.99  996 

40 

20 

8.21  447 

8.21  453 

9.99  994 

40 

30 

8.13  117 

8.13  121 

9.99  996 

30 

30 

8.21  576 

8.21  581 

9.99  994 

30 

40 

8.13  272 

8.13  276 

9.99  996 

20 

40 

8.21  703 

8.21  709 

9.99  994 

20 

50 

8.13  427 

8.13  431 

9.99  996 

10 

50 

8.21  831 

8.21  837 

9.99  994 

10 

470 

8.13  581 

8.13  585 

9.99  996 

013 

57  0 

8.21  958 

8.21  964 

9.99  994 

0  3 

10 

8.13  735 

8.13  739 

9.99  996 

50 

10 

8.22  085 

8.22  091 

9.99  994 

50 

20 

8.13  888 

8.13  892 

9.99  996 

40 

20 

8.22  211 

8.22  217 

9.99  994 

40 

30 

8.14  041 

8.14  045 

9.99  996 

30 

30 

8.22  337 

8.22  343 

9.99  994 

30 

40 

8.14  193 

8.14  197 

9.99  996 

20 

40 

8.22  463 

8.22  469 

9.99  994 

20 

50 

8.14  344 

8.14  348 

9.99  996 

10 

50 

8.22  588 

8.22  595 

9.99  994 

10 

48  0 

8.14  495 

8.14  500 

9.99  996 

012 

58  0 

8.22  713 

8.22  720 

9.99  994 

0  2 

10 

8.14  646 

8.14  650 

9.99  996 

50 

10 

8.22  838 

8.22  844 

9.99  994 

50 

20 

8.14  796 

8.14  800 

9.99  996 

40 

20 

8.22  962 

8.22  968 

9.99  994 

40 

30 

8.14  945 

8.14  950 

9.99  996 

30 

30 

8.23  086 

8.23  092 

9.99  994 

30 

40 

8.15  094 

8.15  099 

9.99  9<;6 

20 

40 

8.23  210 

8.23  216 

9.99  994 

20 

50 

8.15  243 

8.15  247 

9.99  996 

10 

50 

8.23  333 

8.23  339 

9.99  994 

10 

49  0 

8.15  391 

8.15  395 

9.99  996 

Oil 

59  0 

8.23  456 

8.23  462 

9.99  994 

0  1 

10 

8.15  538 

8.15  543 

9.99  996 

50 

10 

8.23  578 

8.23  585 

9.99  994 

50 

20 

8.15  685 

8.15  690 

9.99  996 

40 

20 

8.23  700 

8.23  707 

9.99  994 

40 

30 

8.15  832 

8.15  836 

9.99  996 

30 

30 

8.23  822 

8.23  829 

9.99  993 

30 

40 

8.15  978 

8.15  982 

9.99  995 

20 

40 

8.23  944 

8.23  950 

9.99  993 

20 

50 

8.16  123 

8.16  128 

9.99  995 

10 

50 

8.24  065 

8.24  071 

9.99  993 

10 

50  0 

8.16  268 

8.16  273 

9.99  995 

010 

60  0 

8.24  186 

8.24  192 

9.99  993 

0  0 

/  // 

log  cos 

log  cot 

log  sin 

//  / 

/  // 

log  COS 

log  cot 

log  sin 

//  / 

8 

9°                  1 

25 


Logarithms  of  the  Tkigoxometric  Functions 


1°                  1 

/  // 

log  sin   log  tan 

log  COS 

rr  1 

/  // 

log  sin   log  tan 

log  COS 

//  / 

0  0 

8.24  186  8.24  192 

9.99  993 

060 

10  0 

8.30  879  !  8.30 

888  !  9.99  991 

0  50 

10 

8.24  306  8.24  313 

9.99  993 

50 

10 

8.30  983 

8.30 

992 

9.99  991 

50 

20 

8.24  426  8.24  433 

9.99  993 

40 

20 

8.31  086 

8.31 

095 

9.99  991 

40 

30 

8.24  546  8.24  553 

9.99  993 

30 

30 

8.31  188  8.31 

198 

9.99  991 

30 

40 

8.24  665  8.24  672 

9.99  993 

20 

40 

8.31  291  '  8.31 

300 

9.99  991 

20 

50 

8.24  785 

8.24  791 

9.99  993 

10 

50 

8.31  393  ,  8.31 

403 

9.99  991 

10 

1  0 

8.24  903 

8.24  910 

9.99  993 

0  59 

110 

8.31  495  j  8.31 

505 

9.99  991 

0  49 

10 

8.25  022 

8.25  029 

9.99  993 

50 

10 

8.31  597  1  8.31 

606 

9.99  991 

50 

20 

8.25  140 

8.25  147 

9.99  993 

40 

20 

8.31  699  1  8.31 

708 

9.99  991 

40 

30 

8.25  258  i  8.25  265 

9.99  993 

30 

30 

8.31  800  8.31 

809 

9.99  991 

30 

40 

8.25  375  <  8.25  382 

9.99  993 

20 

40 

8.31  901  8.31 

911 

9.99  991 

20 

50 

8.25  493  8.25  500 

9.99  993 

10 

50 

8.32  002  8.32 

012 

9.99  991 

10 

2  0 

8.25  609  ,  8.25  616 

9.99  993 

0  58 

12  0 

8.32  103  :  8.32 

112 

9.99  990 

0  48 

10 

8.25  726  8.25  733 

9.99  993 

50 

10 

8.32  203  8.32 

213 

9.99  990 

50 

20 

8.25  842  8.25  849 

9.99  993 

40 

20 

8.32  303  8.32 

313 

9.99  990 

40 

30 

8.25  958 

8.25  965 

9.99  993 

30 

30 

8.32  403  8.32 

413 

9.99  990 

30 

40 

8.26  074 

8.26  081 

9.99  993 

20 

40 

8.32  503  i  8.32 

513 

9.99  990 

20 

50 

8.26  189 

8.26  196 

9.99  993 

10 

50 

8.32  602  '  8.32 

612 

9.99  990 

10 

3  0 

8.26  304  8.26  312  | 

9.99  993 

0  57 

13  0 

8.32  702  :  8.32 

711 

9.99  990 

0  47 

10 

8.26  419  8.26  426 

9.99  993 

50 

10 

8.32  801  1  8.32 

811 

9.99  990 

50 

20 

8.26  533 

8.26  541 

9.99  993 

40 

20 

8.32  899- 

8.32 

909 

9.99  990 

40 

30 

8.26  648 

8.26  655 

9.99  993 

30 

30 

8.32  998 

8.33 

008 

9.99  990 

30 

40 

8.26  761 

8.26  769 

9.99  993 

20 

40 

8.33  096 

8.33 

106 

9.99  990 

20 

50 

8.26  875 

8.26  882 

9.99  993 

10 

50 

8.33  195 

8.33 

205 

9.99  990 

10 

4  0 

8.26  988 

8.26  996 

9.99  992 

0  56 

14  0 

8.33  292 

8.33 

302 

9.99  990 

0  46 

10 

8.27  101 

8.27  109 

9.99  992 

50 

10 

8.33  390 

8.33 

400 

9.99  990 

50 

20 

8.27  214 

8.27  221 

9.99  992 

40 

20 

8.33  488 

8.33 

498 

9.99  990 

40 

30 

8.27  326 

8.27  334 

9.99  992 

30 

30 

8.33  585 

8.33 

595 

9.99  990 

30 

40 

8.27  438 

8.27  446 

9.99  992 

20 

40 

8.33  682 

8.33 

692 

9.99  990 

20 

50 

8.27  550 

8.27  558 

9.99  992 

10 

50 

8.33  779 

8.33 

789 

9.99  990 

10 

5  0 

8.27  661 

8.27  669 

9.99  992 

0  55 

15  0 

8.33  875 

8.33 

886 

9.99  990 

0  46 

10 

8.27  773 

8.27  780 

9.99  992 

50 

10 

8.33  972 

8.33 

982 

9.99  990 

50 

20 

8.27  883 

8.27  891 

9.99  992 

40 

20 

8.34  068 

8.34 

078 

9.99  990 

40 

30 

8.27  994 

8.28  002 

9.99  992 

30 

30 

8.34  164 

8.34 

174 

9.99  990 

30 

40  8.28  104 

8.28  112 

9.99  992 

20 

40 

8.34  260 

8.34 

270 

9.99  989 

20 

50 

8.28  215 

8.28  223 

9.99  992 

10 

50 

8.34  355 

8.34 

366 

9.99  989 

10 

6  0 

8.28  324 

8.28  332 

9.99  992 

0  54 

16  0 

8.34  450 

8.34 

461 

9.99  989 

0  44 

10 

8.28  434 

8.28  442 

9.99  992 

50 

10 

8.34  546 

8.34 

556 

9.99  989 

50 

20 

8.28  543 

8.28  551 

9.99  992 

40 

20 

8.34  640 

8.34 

651 

9.99  989 

40 

30 

8.28  652 

8.28  660 

9.99  992 

30 

30 

8.34  735 

8.34 

746 

9.99  989 

30 

40 

8.28  761 

8.28  769 

9.99  992 

20 

40 

8.34  830 

8.34 

840 

9.99  989 

20 

50 

8.28  869 

8.28  877 

9.99  992 

10 

50 

8.34  924 

8.34 

935 

9.99  989 

10 

7  0 

8.28  977 

8.28  986  9.99  992 

0  53 

17  0 

8.35  018 

8.35 

029 

9.99  989 

0  43 

10  8.29  085 

8.29  094 

9.99  992 

50 

10 

8.35  112 

8.35 

123 

9.99  989 

50 

20  1  8.29  193 

8.29  201 

9.99  992 

40 

20 

8.35  206 

8.35 

217 

9.99  989 

40 

30  1  8.29  300 

8.29  309 

9.99  992 

30 

30 

8.35  299 

8.35 

310 

9.99  989 

30 

40  1  8.29  407 

8.29  416 

9.99  992 

20 

40 

8.35  392 

8.35 

403 

9.99  989 

20 

50  8.29  514 

8.29  523 

9.99  992 

10 

50 

8.35  485 

8.35 

497 

9-99  989 

10 

8  0  t  8.29  621 

8.29  629 

9.99  992 

0  52 

18  0 

8.35  578 

8.35 

590 

9.99  989 

0  42 

10  8.29  727 

8.29  736 

9.99  991 

50 

10 

8.35  671 

8.35 

682 

9.99  989 

50 

20  8.29  833 

8.29  842 

9.99  991 

40 

20 

8.35  764 

8.35 

775 

999  989 

40 

30  1  8.29  939 

8.29  947 

9.99  991 

30 

30 

8.35  856 

8.35 

867 

9.99  989 

30 

40  !  8.30  044 

8.30  053 

9.99  991 

20 

40 

8.35  9^8 

8.35 

959 

999  989 

20 

50  :  8.30  150 

8.30  158 

9.99  991 

10 

50 

8.36  040 

8.36 

051  9.99  989 

10 

9  0  1  8.30  255 

8.30  263 

9.99  991 

0  51 

19  0 

8.36  131 

8.36 

143 

9.99  989 

0  41 

10  1  8.30  359 

8.30  368 

9.99  991 

50 

10 

8.36  223 

8.36 

235 

9.99  988 

50 

20  8.30  464 

8.30  473 

9.99  991 

40 

20 

8.36  314 

8.36 

326 

999  988 

40 

30  8.30  568 

8.30  577 

9.99  991 

30 

30 

8.36  405 

8.36 

417 

9.99  988 

30 

40  1  8.30  672 

8.30  681 

9.99  991 

20 

40 

8.36  496 

8.36 

508 

999  988 

20 

50  ;  8.30  776 

8.30  785 

9.99  991 

10 

50 

8.36  587 

8.36 

599 

9.99  988 

10 

10  0  i  8.30  879 

8.30  888 

9.99  991 

0  50 

20  0 

8.36  678 

8.36 

689  9.99  988 

040 

/  // 

log  COS 

log  cot 

log  sin 

'/  / 

/  // 

log  cos   log  cot  j  log  sin  "  '  | 

8$ 

^o 

1 

26 


Logarithms  of  the  Trigonometric  Functions 


1 

1 

/  // 

log 

siu 

log  tail  log  cos 

//  / 

/  // 

log 

sill   log  tan   log  cos 

//  / 

20  0 

8.36 

678 

8.36 

689 

9.99  988 

0  40 

30  0 

8.41 

792  8.41 

807 

9.99  985 

0  30 

10 

8.36 

768 

8.36 

780 

9.99  988 

50 

10 

8.41 

872 

8.41 

887 

9.99  985 

50 

20 

8.36 

858 

8.36 

870 

9.99  988 

40 

20 

8.41 

952 

8.41 

967 

9.99  985 

40 

30 

8.36 

948 

8.36 

960 

9.99  988 

30 

30 

8.42 

032 

8.42 

048 

9.99  985 

30 

40 

8.37 

038 

8.37 

050 

9.99  988 

20 

40 

8.42 

112 

8.42 

127 

9.99  985 

20 

50 

8.37 

128 

8.37 

140 

9.99  988 

10 

50 

8.42 

192 

8.42 

207 

9.99  985 

10 

210 

8.37 

217 

8.37 

229 

9.99  988 

0  39 

310 

8.42 

272 

8.42 

287 

9.99  985 

0  29 

10 

8.37 

306 

8.37 

318 

9.99  988 

50 

10 

8.42 

351 

8.42 

366 

9.99  985 

50 

20 

8.37 

395 

8.37 

408 

9.99  988 

40 

20 

8.42 

430 

8.42 

446 

9.99  985 

40 

30 

8.37 

484 

8.37 

497 

9.99  988 

30 

30 

8.42 

510 

8.42 

525 

9.99  985 

30 

40 

8.37 

573 

8.37 

585 

9.99  988 

20 

40 

8.42 

589 

8.42 

604 

9.99  985 

20 

50 

8.37 

662 

8.37 

674 

9.99  988 

10 

50 

8.42 

667 

8.42 

683 

9.99  985 

10 

22  0 

8.37 

750 

8.37 

762 

9.99  988 

0  38 

32  0 

8.42 

746 

8.42 

762 

9.99  984 

0  28 

10 

8.37 

838 

8.37 

850 

9.99  988 

50 

10 

8.42 

825 

8.42 

840 

9.99  984 

50 

20 

8.37 

926 

8.37 

938 

9.99  988 

40 

20 

8.42 

903 

8.42 

919 

9.99  984 

40 

30 

8.38 

014 

8  38 

026 

9.99  987 

30 

30 

8.42  9S2 

8.42 

997 

9.99  984 

30 

40 

8.38 

101 

8.38 

114 

9.99  987 

20 

40 

8.43 

060 

8.43 

075 

9.99.984 

20 

50 

8.38 

189 

8.38 

202 

9.99  987 

10 

50 

8.43 

138 

8.43 

154 

9.99  984 

10 

23  0 

8.38 

276 

8.38 

289 

9.99  987 

0  37 

33  0 

8.43 

216 

8.43 

232 

9.99  984 

0  27 

10 

8.38 

363 

8.38 

376 

9.99  987 

50 

10 

8.43 

293 

8.43 

309 

9.99  984 

50 

20 

8.38 

450 

8.38 

463 

9.99  987 

40 

20 

8.43 

371 

8.43 

387 

9.99  984 

40 

30 

8.38 

537 

8.38 

550 

9.99  987 

30 

30 

8.43 

448 

8.43 

464 

9.99  984 

30 

40 

8.38 

624 

8.38 

636 

9.99  987 

20 

40 

8.43 

526 

8.43 

542 

9.99  984 

20 

50 

8.38 

710 

8.38 

723 

9.99  987 

10 

50 

8.43 

603 

8.43 

619  1  9.99  984 

10 

24  0 

8.38 

796 

8.38 

809 

9.99  987 

0  36 

34  0 

8.43 

680 

8.43 

696 

9.99  984 

0  26. 

10 

8.38 

882 

8.38 

895 

9.99  987 

50 

10 

8.43 

757 

8.43 

773 

9.99  984 

50 

20 

8.38 

968 

8.38 

981 

9.99  987 

40 

20 

8.43 

834 

8.43 

850 

9.99  984 

40 

30 

8.39 

054 

8.39 

067 

9.99  987 

30 

30 

8.43 

910 

8.43 

927 

9.99  984 

30 

40 

8.39 

139 

8.39 

153 

9.99  987 

20 

40 

8.43 

987 

8.44 

003 

9.99  984 

20 

50 

8:39  225 

8.39 

238 

9.99  987 

10 

50 

8.44 

063 

8.44 

080  i  9.99  983 

10 

25  0 

8.39 

310 

8.39 

323 

9.99  987 

0  35 

35  0 

8.44 

139 

8.44 

156 

9.99  983 

0  26 

10 

8.39 

395 

8.39 

408 

9.99  987 

50 

10 

8.44 

216 

8.44 

232 

9.99  983 

50 

20 

8.39 

480 

8.39 

493 

9.99  987 

40 

20 

8.44 

292 

8.44 

308 

9.99  983 

40 

30 

8.39 

565 

8.39 

578 

9.99  987 

30 

30 

8.44 

367 

8.44 

384 

9.99  983 

30 

40 

8.39  649 

8.39 

663 

9.99  987 

20 

40 

8.44 

443 

8.44 

460 

9.99  983 

20 

50 

8.39 

734 

8.39 

747 

9.99  986 

10 

50 

8.44 

519 

8.44 

536 

9.99  983 

10 

260 

8.39 

818 

8.39 

832 

9.99  986 

034 

360 

8.44 

594 

8.44 

611 

9.99  983 

0  24 

10 

8.39 

902 

8.39 

916 

9.99  986 

50 

10 

8.44 

669 

8.44 

686 

9.99  983 

50 

20 

8.39 

986 

8.40 

000 

9.99  986 

40 

20 

8.44 

745 

8.44 

762 

9.99  983 

40 

30 

8.40 

070 

8.40 

083 

9.99  986 

30 

30 

8.44 

820 

8.44 

837 

9.99  983 

30 

40 

8.40 

153 

8.40 

167 

9.99  986 

20 

40 

8.44 

895 

8.44 

912 

9.99  983 

20 

50 

8.40 

237 

8.40 

251 

9.99  986 

10 

50 

8.44 

969 

8.44 

987 

9.99  983 

10 

27  0 

8.40 

320 

8.40 

334 

9.99  986 

033 

37  0 

8.45 

044 

8.45 

061 

9.99  983 

023 

10 

8.40 

403 

8.40 

417 

9.99  986 

50 

10 

8.45 

119 

8.45 

136 

9.99  983 

50 

20 

8.40 

486 

8.40 

500 

9.99  986 

40 

20 

8.45 

193 

8.45 

210 

9.99  983 

40 

30 

8.40 

569 

8.40 

583 

9.99  986 

30 

30 

8.45 

267 

8.45 

285 

9-99  983 

30 

40 

8.40 

651 

8.40 

665 

9.99  986 

20 

40 

8.45 

341 

8.45 

359 

9.99  982 

20 

50 

8.40 

734 

8.40 

748 

9.99  986 

10 

50 

8.45 

415 

8.45 

433 

9.99  982 

10 

28  0 

8.40 

816 

8.40 

830 

9.99  986 

0  32 

38  0 

8.45 

489 

8.45 

507 

9.99  982 

0  22 

10 

8.40 

898 

8.40 

913 

9.99  986 

50 

10 

8.45 

563 

8.45 

581 

9.99  982 

50 

20 

8.40 

980 

8.40 

995 

9.99  986 

40 

20 

8.45 

637 

8.45 

655 

9.99  982 

40 

30 

8.41 

062 

8.41 

077 

9.99  986 

30 

30 

8.45 

710 

8.45 

728 

9.99  982 

30 

40 

8.41 

144 

8.41 

158 

9.99  986 

20 

40 

8.45 

784 

8.45 

802 

9.99  982 

20 

50 

8.41 

225 

8.41 

240 

9.99  986 

10 

50 

8.45 

857 

8.45 

875 

9.99  982 

10 

290 

8.41 

307 

8.41 

321 

9.99  985 

031 

39  0 

8.45 

930 

8.45 

948 

9.99  982 

021 

10 

8.41 

388 

8.41 

403 

9.99  985 

50 

10 

8.46 

003 

8.46 

021 

9.99  982 

50 

20 

8.41 

469 

8.41 

484 

9.99  985 

40 

20 

8.46 

076 

8.46 

094 

9.99  982 

40 

30 

8.41 

5. SO 

8.41 

565 

9.99  985 

30 

30 

8.46 

149 

8.46 

167 

9.99  982 

30 

40 

8.41 

631 

8.41 

646 

9.99  985 

20 

40 

8.46 

222 

8.46 

240 

9.99  982 

20 

50 

8.41 

711 

8.41 

726 

9.99  985 

10 

50 

8.46 

294 

8.46 

312 

9.99  982 

10 

30  0 

8.41 

792 

8.41 

807 

9.99  985 

0  30 

40  0 

8.46 

366 

8.46 

385 

9.99  982 

0  20 

/  // 

log 

cos 

log  cot   log  sin 

//  / 

/  // 

log 

COS 

log  cot 

log  sin 

//  f 

St 

^o 

1 

97 


Logarithms  of  the  Trigoxometbio  Functions 


1°                 1 

/  // 

log  sin 

log  tan 

log  cos 

II  t 

/  // 

log  sin 

log  tan 

log 

COS 

//  / 

40  0 

8.46  366 

8.46 

385 

9.99  982 

0  20 

50  0 

8.50  504 

8.. SO 

527 

9.99 

978 

010 

10 

8.46  439 

8.46 

457 

9.99  982 

50 

10 

8.50  570 

8.50 

593 

9.99 

978 

50 

20 

8.46  511 

8.46 

529 

9.99  982 

40 

20 

8.50  636 

8. -SO 

658 

9.99 

978 

40 

30 

8.46  583 

8.46 

602 

9.99  981 

30 

30 

8.50  701 

8.50 

724 

9.99 

978 

30 

40 

8.46  655 

8.46 

674 

9.99  981 

20 

40 

8.50  767 

8.50 

789 

9.99 

977 

20 

50 

8.46  727 

8.46 

745 

9.99  981 

10 

50 

8.50  832 

8.50 

855 

9.99 

977 

10 

410 

8.46  799 

8.46 

817 

9.99  981 

019 

510 

8.50  897 

8.50 

920 

9.99 

977 

0  9 

10 

8.46  870 

8.46 

889 

9.99  981 

50 

10 

8.50  963 

8.50 

985 

9.99 

977 

50 

20 

8.46  942 

8.46 

960 

9.99  981 

40 

20 

8.51  028 

8.51 

050 

9.99 

977 

40 

30 

8.47  013 

8.47 

032 

9.99  981 

30 

30 

8.51  092 

8.51 

115 

9.99 

977 

30 

40 

8.47  084 

8.47 

103 

9.99  981 

20 

40 

8.51  157 

8.51 

180 

9.99 

977 

20 

SO 

8.47  155 

8.47 

174 

9.99  981 

10 

50 
52  0 

8.51  222 

8.51 

245 

9.99 

977 

10 

42  0 

8.47  226 

8.47 

245 

9.99  981 

018 

8.51  287 

8.51 

310 

9.99 

977 

0  8 

10 

8.47  297 

8.47 

316 

9.99  981 

50 

10 

8.51  351 

8.51 

374 

9.99  977 

50 

20 

8.47  368 

8.47 

387 

9.99  981 

40 

20 

8.51  416 

8.51 

439 

9.99 

977 

40 

30 

8.47  439 

8.47 

458 

9.99  981 

30 

30 

8.51  480 

8.51 

503 

9.99  977 

30 

40 

8.47  509 

8.47 

528 

9.99  981 

20 

40 

8.51  544 

8.51 

568 

9.99 

977 

20 

50 

8.47  580 

8.47 

599 

9.99  981 

10 

50 

8.51  609 

8.51 

632 

9.99 

977 

10 
0  7 

43  0 

8.47  650 

8.47 

669 

9.99  981 

017 

53  0 

8.51  673 

8.51 

696 

9.99 

977 

10 

8.47  720 

8.47 

740 

9.99  980 

50 

10 

8.51  737 

8.51 

760 

9.99 

976 

50 

20 

8.47  790 

8.47 

810 

9.99  980 

40 

20 

8.51  801 

8.51 

824 

9.99 

976 

40 

30 

8.47  860 

8.47 

880 

9.99  980 

30 

30 

8.51  864 

8.51 

888 

9.99 

976 

30 

40 

8.47  930 

8.47 

950 

9.99  980 

20 

40 

8.51  928 

8.51 

952 

9.99 

976 

20 

50 
44  0 

8.48  000 

8.48 

020 

9.99  980 

10 

50 

8.51  992 

8.52 

015 

9.99 

976 

10 

8.48  069 

8.48 

090 

9.99  980 

016 

54  0 

8.52  055 

8.52 

079 

9.99 

976 

0  6 

10 

8.48  139 

8.48 

159 

9.99  980 

50 

10 

8.52  119 

8.52 

143 

9.99 

976 

50 

20 

8.48  208 

8.48 

228 

9.99  980 

40 

20 

8.52  182 

8.52 

206 

9.99 

976 

40 

30 

8.48  278 

8.48 

298 

9.99  980 

30 

30 

8.52  245 

8.52 

269 

9.99 

976 

30 

40 

8.48  347 

8.48 

367 

9.99  980 

20 

40 

8.52  308 

8.52 

332 

9.99 

976 

20 

50 

8.48  416 

8.48 

436 

9.99  980 

10 

50 

8.52  371 

8.52 

396 

9.99 

976 

10 

45  0 

8.48  485 

8.48 

505 

9.99  980 

015 

55  0 

8.52  434 

8.52 

459 

9.99 

976 

0  5 

10 

8.48  554 

8.48 

574 

9.99  980 

50 

10 

8.52  497 

8.52 

522 

9.99 

976 

50 

20 

8.48  622 

8.48 

643 

9.99  980 

40 

20 

8.52  560 

8.52 

584 

9.99 

976 

40 

30 

8.48  691 

8.48 

711 

9.99  980 

30 

30 

8.52  623 

8.52 

647 

9.99 

975 

30 

40 

8.48  760 

8.48 

780 

9.99  979 

20 

40 

8.52  685 

8.52 

710 

9.99 

975 

20 

50 

8.48  828 

8.48 

849 

9.99  979 

10 

50 

8.52  748 

8.52 

772 

9.99 

975 

10 

46  0 

8.48  896 

8.48 

917 

9.99  979 

014 

560 

8.52  810 

8.52 

835 

9.99 

975 

0  4 

10 

8.48  965 

8.48 

985 

9.99  979 

50 

10 

8.52  872 

8.52 

897 

9.99 

975 

50 

20 

8.49  033 

8.49 

053 

9.99  979 

40 

20 

8.52  935 

8.52 

960 

9.99 

975 

40 

30 

8.49  101 

8.49 

121 

9.99  979 

30 

30 

8.52  997 

8.53 

022 

9.99 

975 

30 

40 

8.49  169 

8.49 

189 

9.99  979 

20 

40 

8.53  059 

8.53 

084 

9.99 

975 

20 

50 

8.49  236 

8.49 

257 

9.99  979 

10 

50  8.53  121 

8.53 

146 

9.99 

975 

10 

47  0 

8.49  304 

8.49 

325 

9.99  979 

013 

57  0 

8.53  183 

8.53 

208 

9.99 

975 

0  3 

10 

8.49  372 

8.49 

393 

9.99  979 

50 

10 

8.53  245 

8.53 

270 

9.99 

975 

50 

20 

8.49  439 

8.49 

460 

9.99  979 

40 

20 

8.53  306 

8.53 

332 

9.99 

975 

40 

30 

8.49  506 

8.49 

528 

9.99  979 

30 

30 

8.53  368 

8.53 

393 

9.99 

975 

30 

40 

8.49  574 

8.49 

595 

9.99  979 

20 

40 

8.53  429 

8.53 

455 

9.99 

975 

20 

50 

8.49  641 

8.49 

662 

9.99  979 

10 

50 

8.53  491 

8.53 

516 

9.99 

974 

10 

48  0 

8.49  708 

8.49 

729 

9.99  979 

012 

58  0 

8.53  552 

8.53 

578 

9.99 

974 

0  2 

10 

8.49  775 

8.49 

796 

9.99  979 

50 

10 

8.53  614 

8.53 

639 

9.99 

974 

50 

20 

8.49  842 

8.49 

863 

9.99  978 

40 

20 

8.53  675 

8.53 

700 

9.99 

974 

40 

30 

8.49  908 

8.49 

930 

9.99  978 

30 

30 

8.53  736 

8.53 

762 

9.99 

974 

30 

40 

8.49  975 

8.49 

997 

9.99  978 

20 

40 

8.53  797 

8.53 

823 

9.99 

974 

20 

50 

8.50  042 

8.50 

063 

9.99  978 

10 

50 

8.53  858 

8.53 

884 

9.99 

974 

10 

49  0 

8.50  108 

8.50 

130 

9.99  978 

Oil 

59  0 

8.53  919 

8.  .S3 

945 

9.99 

974 

0  1 

10 

8.. SO  174 

8.50 

196 

9.99  978 

50 

10 

8.53  979 

8.54 

005 

9.99 

974 

50 

20 

8.50  241 

8.50 

263 

9.99  978 

40 

20 

8.54  040 

8.54 

066 

9.99 

974 

40 

30 

8.50  .307 

8.50 

329 

9.99  978 

30 

30 

8.54  101 

8.54 

127 

9.99 

974 

30 

40 

8.  .50  373 

8.50 

395 

9.99  978 

20 

40 

8.54  161 

8.54 

187 

9.99 

974 

20 

50 

8.50  439 

8.50 

461 

9.99  978 

10 

50 

8.54  222 

8.54 

248 

9.99 

974 

10 

50  0 

8.  .SO  504 

8.50 

527 

9.99  978 

010 

60  0 

8.54  282 

8.54 

308 

9.99 

974 

0  0 

/  // 

log  COS 

log  cot 

log  sin 

//  / 

/  // 

log  cos 

log  cot 

log 

sin 

//  / 

8^ 

^o                             1 

28 


Logarithms  of  the  Trigonometric  Fuxctions 


0°                           *9o= 

180 

^        *  27c 

/ 

log  siu 

d. 

log  I  an 

c.d. 

log  cot 

log  cos 

Prop.  Pts.          1 

0 







0.00  000 

60 

1 

6.46  373 

6.46  373 

3.53  627 

0.00  000 

59 

2 

6.76  476 

30103 

6.76  476 

30103 

3.23  524 

0.00  000 

58 

.3 

6.94  085 

17609 

6.94  085 

17609 

3.05  915 

0.00  000 

57 

4 

7.06  579 

12494 
9691 

7.06  579 

12494 
9691 

2.93  421 

0.00  000 

56 

5 

7.16  270 

7.16  270 

2.83  730 

0.00  000 

55 

6 

7.24  188 

7918 

7.24  188 

7918 

2.75  812 

0.00  000 

54 

7 

7.30  882 

6694 

7.30  8S2 

6694     2.69  118  1 

0.00  000 

53 

8 

7.36  682 

5800 

7.36  682 

5800 

2.63  318 

0.00  000 

52 

9 

7.41  797 

SiiS 

4576 

7.41  797 

511S 
4576 

2.58  203 

0.00  000 

51 

d 

ppl" 

d 

ppl" 

10 

7.46  373 

7.46  373 

2.  .S3  627 

0.00  000 

50 

30103 
17609 

501.72 
293.48 

915 
914 

15-25 
1523 

11 

7.50  512 

4139 

7.50  512 

4139 

2.49  488 

0.00  000 

49 

12494 

208.23 

896 

14-93 

1?, 

7.54  291 

3779 

7.54  291 

3779 

2.45  709 

0.00  000 

48 

9691 

161.52 

89s 

14.92 

13 

7.57  767 

3476 
3218 
2997 

7.57  767 

3476 

2.42  233 

0.00  000 

47 

7918 
6694 

131.97 
111.57 

878 

877 

14.63 
14.62 

14 
15 

7.60  985 

7.60  986 

3219 
2996 

2.39  014 

0.00  000 

46 

5800 
S115 
4576 

96.67 
85-25 
76.27 

860 
843 
878 

14-33 
14.05 
13.80 

7.63  982 

7.63  982 

2.36  018 

0.00  000 

45 

16 

7.66  784 

2802 

7.66  785 

2803 

2.33  215 

0.00  000 

44 

4139 

68.98 

827 

13.78 

17 

7.69  417 

2633 

7.69  418 

2633 

2.30  582 

9.99  999 

43 

3779 

62.98 

812 

13-53 

18 
19 

7.71  900 

7.74  248 

2483 
2348 
2227 

7.71  900 
7.74  248 

2482 
2348 

2.28  100 
2.25  752 

9.99  999 
9.99  999 

42 
41 

3476 
3219 
3218 

57-93 
53-65 
53-63 

797 
782 
769 

13-03 
12.82 

20 

7.76  475 

7.76  476 

2.23  524 

9.99  999 

40 

2997 
2996 
2803 

49-95 
49-93 
46.72 

756 
755 
74s 

12.58 
12.38 

21 

7.78  594 

2119 

7.78  595 

2119 

2.21  405 

9.99  999 

39 

22 

7.80  615 

2021 

7.80  615 

2020 

2.19  385 

9.99  999 

38 

2802 

46.70 

742 

12.37 

23 

7.S2  545 

1930 

7.82  546 

1931 

2.17  454 

9.99  999 

37 

2633 
2483 
2482 

43-88 
41.38 
41-37 

730 

12.17 

24 

7.84  393 

7.84  394 

1848 

2.15  606 

9.99  999 

36 

25 

7.86  166 

7.86  167 

2.13  833 

9.99  999 

35 

2348 

39-13 
37-13 
37-12 

26 

7.87  870 

1704 

7.87  871 

1704 

2.12  129 

9.99  999 

34 

2227 

27 

7.89  509 

1639 

7.89  510 

1639 

2.10  490 

9.99  999 

33 

2119 

35-32 

28 

7.91  088 

1579 

7.91  089 

1579 

2.08  911 

9.99  999 

32 

2021 

33-68 
33-67 
32-18 
32.17 

29 

7.92  612 

1524 
1472 

7.92  613 

1524 
1473 

2.07  387 

9.99  998 

31 
30 

1931 
1930 
1848 
1773 

30 

7.94  084 

7.94  086 

2.05  914 

9.99  998 

31 

7.95  508 

1424 

7.95  510 

1424 

2.04  490 

9.99  998 

29 

29.55 

32 

7.96  887 

1379 

7.96  889 

1379 

2.03  111 

9.99  998 

28 

1704 

28.40 

33 

7.98  223 

1336 

7.98  225 

1336 

2.01  775 

9.99  998 

27 

1639 

27.32 
26.32 
25-40 
24-55 

34 
35 

7.99  520 

1297 
1259 

7.99  522 

1297 
1259 

2.00  478 

9.99  998 

26 

25 

1524 
1473 

8.00  779 

8.00  781 

1.99  219 

9.99  998 

36 

8.02  002 

1223 

8.02  004 

1223 

1.97  996 

9.99  998 

24 

37 

8.03  192 

1 190 

8.03  194 

1 190 

1.96  806 

9.99  997 

23 

1379 

22.98 

38 

8.04  350 

1158 

8.04  353 

1159 

1.95  647 

9.99  997 

22 

1336 

22.27 
21.62 
20.98 
20.38 
19.83 
19-32 
19-30 

39 
40 

8.05  478 

1100 

8.05  481 

1128 

IIOO 

1.94  519 

9.99  997 

21 

1297 
1259 
1223 

8.06  578 

8.06  581 

1.93  419 

9.99  997 

20 

41 
42 

8.07  650 

8.08  696 

1072 
1046 

8.07  653 

8.08  700 

1072 
1047 

1.92  347 
1.91  300 

9.99  997 
9.99  997 

19 
18 

1159 
1158 

43 

8.09  718 

1022 

8.09  722 

1022 

1.90  278 

9.99  997 

17 

1128 

18.80 

44 
45 

8.10  717 

999 
976 

8.10  720 

998 

976 

1.89  280 

9.99  996 

16 

1100 

1072 
1047 

18-33 
17-87 
17-45 

8.11693 

8.11696 

1.88  304 

9.99  996 

15 

46 

8.12  647 

954 

8.12  651 

955 

1.87  349 

9.99  996 

14 

17-43 
17-03 
16.6s 

47 

8.13  581 

934 

8.13  585 

934 

1.86  415 

9.99  996 

13 

999 

48 

8.14  495 

914 

8.14  500 

915 

1.85  500 

9.99  996 

12 

998 

16.63 

49 

8.15  391 

877 

8.15  395 

895 

878 

1.84  605 

9.99  996 

11 

976 
955 

15-92 

50 

8.16  268 

8.16  273 

1.83  727 

9.99  995 

10 

954 

15.90 

51 

8.17  128 

860 

8.17  133 

860 

1.82  867 

9.99  995 

9 

934 

15-57 

52 

8.17  971 

843 

8.17  976 

843 

1.82  024 

9.99  995 

8 

53 

8.18  798 

827 

8.18  804 

828 

1.81  196 

9.99  995 

7 

54 
55 

8.19  610 

797 

8.19  616 

797 

1.80  384 

9.99  995 

6 

8.20  407 

8.20  413 

1.79  587 

9.99  994 

5 

56 

8.21  189 

782 

8.21  195 

782 

1,78  805 

9.99  994 

4 

57 

8.21  958 

769 

8.21  964 

769 

1.78  0.36 

9.99  994 

3 

58 

8.22  713 

755 

8.22  720 

756 

1.77  280 

9.99  994 

2 

59 

8.23  456 

743 

8.23  462 

742 
730 

1.76  538 

9.99  994 

1 

60 

8.24  186 

8.24  192 

1.75  808 

9.99  993 

0 

log  cos 

d. 

log  cot 

C.d. 

log  tan 

log  sin 

1 

Prop.  Pts.         1 

*I79° 

269° 

*359" 

89° 

1 

29 


Logarithms  of  the  Trigonometric  Functions 


»9i'- 


i8i° 


*2-Jl'^ 


lojr  sin      d. 


log  tan  \  c.  d 


log  cot        log  COS 


Prop.  Pts. 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


40 

41 

42 
43 
44 


55 
56 
57 
58 
59 
60 


8.24 
8.24 
8.25 
8.26 
8.26 


186 
903 
609 
304 
988 


8.27 
8.28 
8.28 
8.29 
8.30 


661 
324 
977 
621 

255 


8.30 
8.31 
8.32 
8.32 
8.33 


879 
495 
103 
702 
292 


8.33 
8.34 
8.35 
8.35 
8.36 


875 
450 
018 
578 
131 


8.36 
8.37 
8.37 
8.38 
8.38 


678 
217 
750 
276 
796 


8.39 
8.39 
8.40 
8.40 
8.41 


310 
818 
320 
816 
307 


8.41 
8.42 
8.42 
8.43 
8.43 


792 
272 
746 
216 
680 


8.44 
8.44 
8.45 
8.45 
8.45 


139 
594 
044 
489 
930 


8.46 
8.46 
8.47 
8.47 
8.48 


366 
799 
226 
650 
069 


8.48 
8.48 
8.49 
8.49 
8.50 


485 
896 
304 
708 
108 


8.50 
8.50 
8.51 
8.51 
8.52 


504 
897 
287 
673 
055 


8.52 
8.52 
8.53 
8.53 
8.53 


434 
810 
183 
552 
919 


8.54  282 
log  cos 


717 

706 

69s 
684 
673 

663 

6S3 
644 
634 
624 

616 
608 
599 
590 
S83 

57S 
568 
560 
553 
547 

539 
533 
526 
520 
514 
508 
502 
496 
491 
48s 
480 
474 
470 
464 
459 

455 
450 
445 

441 

436 

433 
427 
424 
419 
416 

411 
408 
404 
400 
396 

393 
390 
386 
382 
379 

376 
373 
369 
367 
363 


8.24  192 

8.24  910 

8.25  616 

8.26  312 
8.26  996 


8.27  669 

8.28  332 

8.28  986 

8.29  629 

8.30  263 


8.30  888 

8.31  505 

8.32  112 

8.32  711 

8.33  302 


8.33  886 

8.34  461 

8.35  029 

8.35  590 

8.36  143 


8.36  689 

8.37  229 

8.37  762 

8.38  289 
8.38  809 


8.39  323 

8.39  832 

8.40  334 

8.40  830 

8.41  321 


8.41  807 

8.42  287 

8.42  762 

8.43  232 
8.43  696 


.44  156 
.44  611 
.45  061 
.45  507 
.45  948 


.46  385 
.46  817 
.47  245 
.47  669 
.48  089 


8.48  505 

8.48  917 

8.49  325 

8.49  729 

8.50  130 


8.50  527 

8.50  920 

8.51  310 

8.51  696 

8.52  079 


»i78°   268° 


8.52  459 

8.52  835 

8.53  208 
8.53  578 

8.53  945 

8.54  308 

log  cot 

*358° 


718 
706 
696 
684 
673 

663 

654 
643 
634 
62s 

617 
607 

599 
591 

584 

575 
568 
S6i 
553 
546 

540 
533 
527 
520 
514 

509 
502 
496 
491 
486 

480 
475 
470 
464 
460 

455 
450 
446 
441 
437 

432 
428 
424 
420 
416 

412 
408 
404 
401 
397 

393 
390 

386 
383 
380 

376 
373 
370 
367 
363 

c!T 


1.75  808 

1.75  090 

1.74  384 

1.73  688 

1.73  004 


9.99  993 
9.99  993 
9.99  993 
9.99  993 
9.99  992 


1.72  331 

1.71  668 

1.71  014 

1.70  371 

1.69  737 


9.99  992 
9.99  992 
9.99  992 
9.99  992 
9.99  99] 


1.69  112 
1.68  495 
1.67  888 
1.67  289 
1.66  698 


9.99  991 
9.99  991 
9.99  990 
9.99  990 
9.99  990 


1.66  114 

1.65  539 

1.64  971 

1.64  410 

1.63  857 


9.99  990 
9.99  989 
9.99  989 
9.99  989 
9.99  989 


1.63  311 

1.62  771 

1.62  238 

1.61  711 

1.61  191 


9.99  988 
9.99  988 
9.99  988 
9.99  987 
9.99  987 


1.60  677 
1.60  168 
1.59  666 
1.59  170 
1.58  679 


9.99  987 
9.99  986 
9.99  986 
9.99  986 
9.99  985 


1.58  193 

1.57  713 

1.57  238 

1.56  768 

1.56  304 


9.99  985 
9.99  985 
9.99  984 
9.99  984 
9.99  984 


1.55  844 

1.55  389 

1.54  939 

1..54  493 

1.54  052 


9.99  983 
9.99  983 
9.99  983 
9.99  982 
9.99  982 


1.53  615 

1.53  183 

1.52  755 

1.52  331 

1.51  911 


9.99  982 
9.99  981 
9.99  981 
9.99  981 
9.99  980 


1.51  495 
1.51  083 
1.50  675 
1.50  271 
1.49  870 


9.99  980 
9.99  979 
9.99  979 
9.99  979 
9.99  978 


1.49  473 

1.49  080 

1.48  690 

1.48  304 

1.47  921 


9.99  978 
9.99  977 
9.99  977 
9.99  977 
9.99  976 


1.47  541 
1.47  165 
1.46  792 
1.46  422 
1.46  055 
1.45  692 
log  tan 

"88^ 


9.99  976 
9.99  975 
9.99  975 
9.99  974 
9.99  974 
9.99  974 
log  sin 


718 
717 
706 
696 
69s 
684 
673 
663 
654 
653 
644 

643 
634 
625 
624 
617 
616 
608 
607 
599 
S91 
590 
S84 
583 
575 
568 
561 
560 

553 
547 
546 
540 
539 
533 
527 
526 
520 
514 
509 
508 
502 
496 
491 
486 


ppl" 

d 

11.97 

48s 

11-95 

480 

11.77 

475 

11.60 

474 

II. S8 

470 

11.40 

464 

11.22 

460 

11.05 

459 

10.90 

455 

10.88 

450 

10.73 

446 

10.72 

445 

IO.S7 

441 

10.42 

437 

10.40 

436 

10.28 

433 

10.27 

432 

10.13 

428 

10.12 

427 

9.98 

424 

9.8s 

420 

9.83 

419 

9-73 

416 

9.72 

412 

9.58 

411 

9-47 

408 

9-35 

404 

9-33 

401 

9.22 

4oo 

9.12 

397 

9.10 

396 

9.00 

393 

8.98 

390 

8.88 

386 

8.78 

383 

8.77 

382 

8.67 

380 

8.57 

379 

8.48 

376 

8.47 

373 

8.37 

370 

8.27 

369 

8.18 

367 

8.10 

363 

ppl" 

8.08 
8.00 
7.92 
7.90 

7.83 

7.73 
7.67 
7.65 

7.58 

7.50 

7-43 
7.42 
7-35 
7.28 
7.27 
7.22 
7.20 
7-13 
7.12 
7.07 
7.00 
6.98 
6.93 
6.87 
6.8s 
6.80 
6.73 
6.68 
6.67 
6.62 
6.60 
6.55 
6.50 
6.43 
6.38 
6.37 
6.33 
6.32 
6.27 
6.22 
6.17 
6.IS 
6.12 
6.05 


Prop.  Pts. 


30 


Logarithms  of  the  Trigonometric  Functions 


2°                           *92 

182=^            *272' 

t 

log  sill 

d. 

log  tan 

c.d. 

log  cot 

log  COS 

Prop.  Pts. 

0 

8.54  282 

8.54  308 

1.45  692 

9.99  974 

60 

1 

8.54  642 

360 

8.54  669 

361 

1.45  331 

9.99  973 

59 

2 

8.54  999 

357 

8.55  027 

358 

1.44  973 

9.99  973 

58 

3 

8.55  354 

355 

8.55  382 

355 

1.44  618 

9.99  972 

57 

4 

8.55  705 

3SI 
349 

8.55  734 

3S2 
349 

1.44  266 

9.99  972 

56 

5 

8.56  054 

8.56  083 

1.43  917 

9.99  971 

55 

6 

8.56  400 

346 

8.56  429 

346 

1.43  571 

9.99  971 

54 

7 

8.56  743 

343 

8.56  773 

344 

1.43  227 

9.99  970 

53 

8 

8.57  084 

341 

8.57  114 

341 

1.42  886 

9.99  970 

52 

9 

8.57  421 

337 
336 

8.57  452 

338 
336 

1.42  548 

9.99  969 

51 

10 

8.57  757 

8.57  788 

1.42  212 

9.99  969 

50 

11 

8.58  089 

332 

8.58  121 

333 

1.41  879 

9.99  968 

49 

12 

8.58  419 

330 

8.58  451 

330 

1.41  549 

9.99  968 

48 

13 

8.58  747 

328 

8.58  779 

328 

1.41  221 

9.99  967 

47 

14 

8.  .59  072 

32s 

8.59  105 

326 

1.40  895 

9.99  967 

46 

d 

361 

ppi 

6.02 

291 

PPl' 

4.8s 

15 

8.59  395 

8.59  428 

1.40  572 

9.99  967 

45 

360 

6.00 

290 

4.83 

16 

8.59  715 

320 

8.59  749 

321 

1.40  251 

9.99  966 

44 

358 

5.97 

289 

288 
287 

4.82 
4.80 
4.78 

17 

8.60  033 

318 

8.60  068 

319 

1.39  932 

9.99  966 

43 

355 

5.02 

18 

8.60  349 

316 

8.60  384 

316 

1.39  616 

9.99  965 

42 

352 

S.87 

285 

4.75 

19 

8.60  662 

313 

8.60  698 

314 

1.39  302 

9.99  964 

41 

351 

S.8S 
5.82 

284 
78^ 

4.73 

20 

8.60  973 

8.61  009 

311 

1.38  991 

9.99  964 

40 

346 

5. 77 

281 

4.68 

21 

8.61  282 

3og 

8.61  319 

310 

1.38  681 

9.99  963 

39 

344 

5-73 

280 

4.67 
4.65 
4.63 

22 

8.61  589 

307 

8.61  626 

307 

1.38  374 

9.99  963 

38 

341 

5.68 

■>!% 

23 

8.61  894 

305 

8.61  931 

30s 

1.38  069 

9.99  962 

37 

338 

5.63 

277 

4.62 

?4 

8.62  196 

302 

8.62  234 

303 

1.37  766 

9.99  962 

36 

337 

5.62 

276 

4.60 

301 

301 

336 
333 

5.<X3 

5.55 

274 
273 

4.57 
4.55 

25 

8.62  497 

8.62  535 

1.37  465 

9.99  961 

35 

26 

8.62  795 

298 

8.62  834 

299 

1.37  166 

9.99  961 

34 

332 

5.53 

272 

4.53 

27 

8.63  091 

296 

8.63  131 

297 

1.36  869 

9.99  960 

33 

330 
328 

5.50 

271 

4.52 
4.50 
4.48 

28 

8.63  385 

294 

8.63  426 

29s 

1.36  574 

9.99  960 

32 

326 

5.43 

260 

29 

8.63  678 

293 

8.63  718 

292 

1.36  282 

9.99  959 

31 

32s 
323 
321 

5.42 
5.38 
5.35 

268 
267 
266 

4.47 
4.45 
4.43 

30 

8.63  968 

8.64  009 

1.35  991 

9.99  959 

30 

31 

8.64  256 

288 

8.64  298 

289 

1.35  702 

9.99  958 

29 

320 

5-33 

264 

4.40 

32 
33 

8.64  543 
8.64  827 

287 
284 

8.64  585 
8.64  870 

287 
285 

1.35  415 
1.35  130 

9.99  958 
9.99  957 

28 
27 

319 
318 
316 

5.32 
5.30 
5.27 

263 
261 
260 

4.38 
4.35 
4.33 

34 

8.65  110 

283 
281 

8.65  154 

284 
281 

1.34  846 

9.99  956 

26 

314 

5-23 

259 
258 

2';7 

4.32 

35 

8.65  391 

8.65  435 

1.34  565 

9.99  956 

25 

311 

■!.i8 

4.28 

36 

8.65  670 

279 

8.65  715 

280 

1.34  285 

9.99  955 

24 

310 

5-17 

2S6 

4.27 

37 

8.65  947 

277 

8.65  993 

278 

1.34  007 

9.99  955 

23 

309 

5-15 

255 

4.25 

38 

8.66  223 

276 

8.66  269 

276 

1.33  731 

9.99  954 

22 

307 
305 

5-12 

5.08 

264 

253 

4.23 
4.22 

39 

8.66  497 

274 
272 

8.66  543 

274 
273 

1.33  457 

9.99  954 

21 

303 
302 
301 

5-°5 
5-03 
5.02 

252 
251 
250 

4.20 
4.18 
4.17 

40 

8.66  769 

8.66  816 

1.33  184 

9.99  953 

20 

41 

8.67  039 

270 

8.67  087 

271 

1.32  913 

9.99  952 

19 

299 

4.98 

249 

4.15 

42 

8.67  308 

269 

8.67  356 

269 

1.32  644 

9.99  952 

18 

298 

4-97 

248 

4.13 

43 

8.67  575 

267 

8.67  624 

268 

1.32  376 

9.99  951 

17 

296 

246 

4.10 

44 

8.67  841 

266 
263 

8.67  890 

266 
264 

1.32  110 

9.99  951 

16 

29s 
294 
293 

4.92 
4.90 
4.88 

245 
244 
24'i 

4.08 
4.07 
4.05 

45 

8.68  104 

8.68  154 

1.31  846 

9.99  950 

15 

46 

8.68  367 

263 

8.68  417 

263 

1.31  583 

9.99  949 

14 

292 

4.87 

242 

4.03 

47 

8.68  627 

260 

8.68  678 

261 

1.31  322 

9.99  949 

13 

48 

8.68  886 

259 

8.68  938 

260 

1.31  062 

9.99  948 

12 

49 

8.69  144 

258 
256 

8.69  196 

258 
257 

1.30  804 

9.99  948 

11 

50 

8.69  400 

8.69  453 

1.30  547 

9.99  947 

10 

51 

8.69  654 

254 

8.69  708 

2SS 

1.30  292 

9.99  946 

9 

52 

8.69  907 

253 

8.69  962 

254 

1.30  038 

9.99  946 

8 

53 

8.70  159 

252 

8.70  214 

252 

1.29  786 

9.99  945 

7 

54 

8.70  409 

250 
249 

8.70  465 

251 
249 

1.29  535 

9.99  944 

6 

55 

8.70  658 

8.70  714 

1.29  286 

9.99  944 

5 

56 

8.70  90S 

247 

8.70  962 

248 

1.29  -038 

9.99  943 

4 

57 

8.71  151 

246 

8.71  208 

246 

1.28  792 

9.99  942 

3 

58 

8.71  395 

244 

8.71  453 

245 

1.28  547 

9.99  942 

2 

59 

8.71  638 

243 
242 

8.71  697 

244 
243 

1.28  303 

9.99  941 

1 

60 

8.71  880 

8.71  940 

1.28  060 

9.99  940 

0 

log  COS 

d. 

log  cot 

C.d. 

log  tan 

log  sin 

/ 

Prop.  Pts. 

*i77° 

267° 

*357° 

87° 

31 


Logarithms  of  the  Trigonometric  Functions 


3°                      *9f 

183"          *27. 

'       1 

1 

log  sin 

(1. 

log  tan 

c.d. 

log  cot 

log  cos 

Prop.  Pts.          1 

0 

8.71  880 

8.71  9^0 

1.28  060 

9.99  940 

60 

1 

8.72  120 

240 

8.72  181 

241 

1.27  819 

9.99  940 

59 

2 

8.72  359 

239 

8.72  420 

239 

1.27  580 

9.99  939 

58 

3 

8.72  597 

238 

8.72  659 

239 

1.27  341 

9.99  938 

57 

4 

8.72  834 

237 
23s 

8.72  896 

237 
236 

1.27  104 

9.99  938 

56 

5 

8.73  069 

8.73  132 

1.26  868 

9.99  937 

55 

6 

8.73  303 

234 

8.73  366 

234 

1.26  634 

9.99  936 

54 

7 

8.73  535 

232 

8.73  600 

234 

1.26  400 

9.99  936 

53 

8 

8.73  767 

232 

8.73  832 

232 

1.26  168 

9.99  935 

52 

9 

8.73  997 

230 

22g 

8.74  063 

231 
220 

1.25  937 

9.99  934 

51 

10 

8.74  226 

8.74  292 

1.25  708 

9.99  934 

50 

11 

8.74  454 

228 

8.74  521 

229 

1.25  479 

9.99  933 

49 

12 

8.74  680 

226 

8.74  748 

227 

1.25  252 

9.99  932 

48 

13 

8.74  906 

226 

8.74  974 

226 

1-25  026 

9.99  932 

47 

14 
15 

8.75  130 

224 
223 

8.75  199 

223 
224 

1.24  801 

9.99  931 

46 

8.75  353 

8.75  423 

1.24  577 

9.99  930 

45 

16 

8.75  575 

222 

8.75  645 

222 

1.24  355 

9.99  929 

44 

17 

8.75  795 

220 

8.75  867 

222 

1.24  133 

9.99  929 

43 

d 

PPI" 

d 

ppl" 

18 

8.76  015 

220 

8.76  087 

220 

1.23  913 

9.99  928 

42 

241 

4.02 

209 

3.48 

19 

8.76  234 

219 

8.76  306 

219 

1.23  694 

9.99  927 

41 

240 

4.00 
3.98 

208 

3.47 

217 

20 

8.76  451 

8.76  525 

1.23  475 

9.99  926 

40 

238 

3-97 

206 

3-43 

21 

8.76  667 

216 

8.76  742 

217 

1.23  258 

9.99  926 

39 

237 
236 
235 

3-95 

20s 

3-42 

22 

8.76  883 

216 

8.76  958 

216 

1.23  042 

9.99  925 

38 

3-93 
392 

203 

3.38 

23 

8.77  097 

214 

8.77  173 

21S 

1.22  827 

9.99  924 

37 

234 

3.90 

202 

3.37 

24 

8.77  310 

213 

8.77  387 

214 

1.22  613 

9.99  923 

36 

233 

3.88 
3.87 

201 

3.3s 

25 

8.77  522 

8.77  600 

1.22  400 

9.99  923 

35 

231 

3.8.-; 

199 

3.32 

26 

8.77  733 

211 

8.77  811 

211 

1.22  189 

9.99  922 

34 

230 

3.83 
3.82 
3.80 

i()8 

3.30 
3.28 
3.27 

27 

8.77  943 

210 

8.78  022 

211 

1.21  978 

9.99  921 

33 

228 

197 

28 

8.78  152 

209 

8.78  232 

210 

1.21  768 

9.99  920 

32 

227 

.V78 

19'; 

3.2s 

29 

8.78  360 

8.78  441 

209 

1.21  559 

9.99  920 

31 

22() 

3.77 

194 

3.23 

208 

3.75 
3-73 

193 
192 

3.22 
3.20 

30 

8.78  568 

8.78  649 

1.21  351 

9.99  919 

30 

224 

31 

8.78  774 

206 

8.78  855 

206 

1.21  145 

9.99  918 

29 

223 

3-72 

191 

3.18 

32 

8.78  979 

20s 

8.79  061 

206 

1.20  939 

9.99  917 

28 

222 

3-70 
3.68 
^.67 

190 
189 
188 

3.17 
3-15 
3.13 

33 

8.79  183 

204 

8.79  266 

20s 

1.20  734 

9.99  917 

27 

220 

34 

8.79  386 

203 

8.79  470 

204 

1.20  530 

9.99  916 

26 

219 
218 
217 

3.6s 
3.63 
3.62 

187 
186 

i8s 

3.12 

35 

8.79  588 

8.79  673 

1.20  327 

9.99  915 

25 

3.08 

36 

8.79  789 

201 

8.79  875 

202 

1.20  125 

9.99  914 

24 

216 

3.60 

184 

3.07 

37 

8.79  990 

201 

8.80  076 

201 

1.19  924 

9.99  913 

23 

215 

3.58 

183 
182 
t8t 

3.05 

38 

8.80  189 

199 

8.80  277 

201 

1.19  723 

9.99  913 

22 

213 

3-55 

3.02 

39 

8.80  388 

199 
197 

8.80  476 

199 
198 

1.19  524 

9.99  912 

21 

212 

3-53 

40 

8.80  585 

8.80  674 

1.19  326 

9.99  911 

20 

210 

3.52 

41 

8.80  782 

197 

8.80  872 

198 

1.19  128 

9.99  910 

19 

42 

8.80  978 

196 

8.81  068 

196 

I.IS  932 

9.99  909 

18 

43 

8.81  173 

195 

8.81  264 

196 

1.18  736 

9.99  909 

17 

44 

8.81  367 

194 
193 

8.81  459 

195 
194 

1.18  541 

9.99  908 

16 

45 

8.81  560 

8.81  653 

1.18  347 

9.99  907 

15 

46 

8.81  752 

192 

8.81  846 

193 

1.18  154 

9.99  906 

14 

47 

8.81  944 

192 

8.82  038 

192 

1.17  962 

9.99  905 

13 

48 

8.82  134 

190 

8.82  230 

192 

1.17  770 

9.99  904 

12 

49 

8.82  324 

190 
189 

8.82  420 

190 
190 

1.17  580 

9.99  904 

11 

50 

8.82  513 

8.82  610 

1.17  390 

9.99  903 

10 

51 

8.82  701 

188 

8.82  799 

189 

1.17  201 

9.99  902 

9 

52 

8.82  888 

187 

8.82  987 

188 

1.17  013 

9.99  901 

8 

53 

8.83  075 

187 

8.83  175 

188 

1.16  825 

9.99  900 

7 

54 

8.83  261 

186 
185 

8.83  361 

186 
186 

1.16  639 

9.99  899 

6 

55 

8.83  446 

8.83  547 

1.16  453 

9.99  898 

5 

56 

8.83  630 

184 

8.83  732 

18s 

1.16  268 

9.99  898 

4 

57 

8.83  813 

183 

8.83  916 

184 

1.16  084 

9.99  897 

3 

58 

8.83  996 

183 

8.84  100 

184 

1.15  900 

9.99  896 

2 

59 

8.84  177 

181 
181 

8.84  282 

182 
182 

1.15  718 

9.99  895 

1 

60 

8.84  358 

8.84  464 

1.15  536 

9.99  894 

0 

log  cos 

d. 

log  cot 

c.d. 

log  tan 

log  sin 

'  1          Prop.  Pts.         1 

*i76° 

266° 

*356° 

86° 

1 

32 


Logarithms  of  the  Trigonometric  Functions 


4°                        *94= 

184°         *274'^ 

0 

log  sin 

d. 

log  tan 

c.d. 

log  cot 

log  COS 

Prop.  Pts. 

8.84  358 

8.84  464 

1.15  536 

9.99  894 

60 

1 

8.84  539 

i8i 

8.84  646 

182 

1.15  354 

9.99  893 

59 

2 

8.84  718 

179 

8.84  826 

180 

1.15  174 

9.99  892 

58 

3 

8.84  897 

179 

8.85  006 

1.14  994 

9.99  891 

57 

4 

5 

8.85  075 

178 
177 

8.85   185 

179 
178 

1.14  815 

9.99  891 

56 

8.85  252 

8.85  363 

1.14  637 

9.99  890 

55 

6 

8.85  429 

177 

8.85  540 

177 

1.14  460 

9.99  889 

54 

7 

8.85  605 

176 

8.85  717 

177 

1.14  283 

9.99  888 

53 

8 

8.85  780 

I7S 

8.85  893 

176 

1.14  107 

9.99  887 

52 

9 
10 

8.85  955 

175 
173 

8.86  069 

176 
174 

1.13  931 

9.99  886 

51 

8.86  128 

8.86  243 

1.13  757 

9.99  885 

50 

11 

8.86  301 

173 

8.86  417 

174 

1.13  583 

9.99  884 

49 

12 

8.86  474 

173 

8.86  591 

174 

1.13  409 

9.99  883 

48 

13 

8.86  645 

171 

8.86  763 

172 

1.13  237 

9.99  882 

47 

14 

8.86  816 

171 
171 

8.86  935 

172 
171 

1.13  065 

9.99  881 

46 

15 

8.86  987 

8.87  106 

1.12  894 

9.99  880 

45 

16 

8.87  156 

169 

8.87  277 

171 

1.12  723 

9.99  879 

44 

d 

PPl" 

17 

8.87  325 

169 

8.87  447 

170 

1.12  553 

9.99  879 

43 

182 

3.03 

18 

8.87  494 

169 

8.87  616 

169 

1.12  384 

9.99  878 

42 

3.02 

19 

8.87  661 

167 
i68 

8.87  785 

169 
168 

1.12  215 

9.99  877 

41 

179 
178 
177 
176 

2.98 
2.97 

20 

8.87  829 

8.87  953 

1.12  047 

9.99  876 

40 

21 

8.87  995 

166 

8.88  120 

167 

1.11  880 

9.99  875 

39 

2.93 

22 

8.88  161 

166 

8.88  287 

167 

1.11  713 

9.99  874 

38 

I7S 

2.92 

23 

8.88  326 

165 

8.88  453 

1 56 

1.11   547 

9.99  873 

37 

174 

2.90 
2.88 
2.87 
2.8s 
2.83 
2.82 

24 

8.88  490 

164 
164 

8.88  618 

16s 
i6s 

1.11  382 

9.99  872 

36 

172 
171 

25 

8.88  654 

8.88  783 

1.11  217 

9.99  871 

35 

26 

8.88  817 

163 

8.88  948 

i6s 

1.11  052 

9.99  870 

34 

169 

27 

8.88  980 

163 

8.89  111 

163 

1.10  889 

9.99  869 

33 

168 

2.80 

28 

8.89  142 

162 

8.89  274 

163 

1.10  726 

9.99  868 

32 

167 
166 
i6s 
164 
163 

2.78 

29 
30 

8.89  304 

162 
160 

8.89  437 

163 
161 

1.10  563 

9.99  867 

31 

2.75 
2.73 

8.89  464 

8.89  598 

1.10  402 

9.99  866 

30 

31 

8.89  625 

161 

8.89  760 

162 

1.10  240 

9.99  865 

29 

32 

8.89  784 

159 

8.89  920 

160 

1.10  080 

9.99  864 

28 

161 

2.68 

33 

8.89  943 

159 

8.90  080 

160 

1.09  920 

9.99  863 

27 

160 

2.67 

34 

8.90  102 

159 
IS8 

8.90  240 

160 
159 

1.09  760 

9.99  862 

26 

159 

IS8 
157 

2.63 
2.62 

35 

8.90  260 

8.90  399 

1.09  601 

9.99  861 

25 

36 

8.90  417 

157 

8.90  557 

158 

1.09  443 

9.99  860 

24 

156 
155 
154 

2.58 
2.57 

37 

8.90  574 

157 

8.90  715 

158 

1.09  285 

9.99  859 

23 

38 

8.90  730 

156 

8.90  872 

157 

1.09  128 

9.99  858 

22 

153 

2.5S 

39 

8.90  885 

155 
ISS 

8.91  029 

157 
156 

1.08  971 

9.99  857 

21 

152 
iSi 
ISO 

2.53 
2.52 
2.50 

40 

8.91  040 

8.91  185 

1.08  815 

9.99  856 

20 

41 
42 

8.91  195 
8.91  349 

155 

154 

8.91  340 
8.91  495 

ISS 

ISS 

1.08  660 
1.08  505 

9.99  855 
9.99  854 

19 

18 

149 

148 
147 

2.47 
2.4s 

43 

8.91  502 

153 

8.91  650 

ISS 

1.08  350 

9.99  853 

17 

146 

2.43 

44 

8.91  655 

153 
152 

8.91  803 

IS3 
IS4 

1.08  197 

9.99  852 

16 

145 

2.42 

45 

8.91  807 

8.91  957 

1.08  043 

9.99  851 

15 

46 

8.91  959 

152 

8.92  110 

153 

1.07  890 

9.99  850 

14 

47 

8.92  110 

151 

8.92  262 

152 

1.07  738 

9.99  848 

13 

48 

8.92  261 

151 

8.92  414 

152 

1.07  586 

9.99  847 

12 

49 
50 

8.92  411 

150 
150 

8.92  565 

151 
151 

1.07  435 

9.99  846 

11 

8.92  561 

8.92  716 

1.07  284 

9.99  845 

10 

51 

8.92  710 

149 

8.92  866 

ISO 

1.07  134 

9.99  844 

9 

52 

8.92  859 

149 

8.93  016 

150 

1.06  984 

9.99  843 

8 

53 

8.93  007 

148 

8.93  165 

149 

1.06  835 

9.99  842 

7 

54 

55 

8.93  154 

147 
147 

8.93  313 

149 

1.06  687 

9.99  841 

6 

5 

8.93  301 

8.93  462 

1.06  538 

9.99  840 

56 

8.93  448 

147 

8.93  609 

147 

1.06  391 

9.99  839 

4 

57 

8.93  594 

14b 

8.93  756 

147 

1.06  ?.44 

9.99  838 

3 

58 

8.93  740 

146 

8.93  903 

147 

1.06  097 

9.99  837 

2 

59 
60 

8.93  885 

145 
145 

8.94  049 

146 
146 

1.05  951 

9.99  836 

1 

8.94  030 

8.94  195 

1.05  805 

9.99  834 

0 

log  cos 

d. 

log  cot 

C.d. 

log  tan 

log  sin 

/ 

Prop.  Pts. 

*i75° 

265^ 

*355° 

85° 

Logarithms  of  the  Tbigokometric  Functions 


5°                            *95 

°        185=        *275- 

"o 

log  sin 

d. 

log  tan 

c.d. 

log  cot 

log  cos 

Prop.  Pts. 

8.94  030 

8.94  195 

1.05  805 

9.99  834 

60 

1 

8.94  174 

144 

8.94  340 

14s 

1.05  660 

9.99  833 

59 

2 

8.94  317 

143 

8.94  485 

145 

1.05  515 

9.99  832 

58 

3 

8.94  461 

144 

8.94  630 

14s 

1.05  370 

9.99  831 

57 

4 

8.94  603 

143 

8.94  773 

144 

1.05  227 

9.99  830 

56 

5 

8.94  746 

8.94  917 

1.05  083 

9.99  829 

55 

6 

8.94  887 

I4i 

8.95  060 

143 

1.04  940 

9.99  828 

54 

7 

8.95  029 

142 

8.95  202 

142 

1.04  798 

9.99  827 

53 

8 

8.95  170 

141 

8.95  344 

142 

1.04  656 

9.99  825 

52 

9 
10 

8.95  310 

140 

8.95  486 

141 

1.04  514 

9.99  824 

51 

8.95  450 

8.95  627 

1.04  373 

9.99  823 

50 

11 

8.95  589 

139 

8.95  767 

140 

1.04  233 

9.99  822 

49 

12 

8.95  728 

139 

8.95  908 

141 

1.04  092 

9.99  821 

48 

13 

8.95  867 

138 
138 

8.96  047 

139 

1.03  953 

9.99  820 

47 

14 

8.96  005 

8.96  187 

138 

1.03  813 

9.99  819 

46 

15 

8.%  143 

8.%  325 

1.03  675 

9.99  817 

45 

16 

8.96  280 

137 

8.96  464 

139 

1.03  536 

9.99  816 

44 

17 

8.%  417 

137 
136 
136 
136 

8.96  602 

138 

1.03  398 

9.99  815 

43 

18 

8.%  553 

8.%  739 

137 
138 
136 

1.03  261 

9.99  814 

42 

19 

8.96  689 

8.96  877 

1.03  123 

9.99  813 

41 

20 

8,96  825 

8.97  013 

1.02  987 

9.99  812 

40 

d 

ppi" 

21 

8.96  960 

135 

8.97  150 

137 

1.02  850 

9.99  810 

39 

22 

8.97  095 

13s 

8.97  285 

I3S 
136 

1.02  715 

9.99  809 

38 

14s 

2.42 

23 

8.97  229 

134 

8.97  421 

1.02  579 

9.99  808 

37 

143 

2.38 

24 

8.97  363 

133 

8.97  556 

135 

1.02  444 

9.99  807 

36 

142 
141 

2.37 

2.35 

25 

8.97  496 

8.97  691 

1.02  309 

9.99  806 

35 

26 

8.97  629 

133 

8.97  825 

134 

1.02  175 

9.99  804 

34 

139 

2.32 

27 

8.97  762 

133 

8.97  959 

134 

1.02  041 

9.99  803 

33 

138 

2.30 

28 

8.97  894 

132 

8.98  092 

133 

1.01  908 

9.99  802 

32 

137 

n6 

29 

8.98  026 

131 

S.98  225 

133 

1.01  775 

9.99  801 

31 

13s 
134 
133 
132 

2.2s 
2.23 

30 

8.98  157 

8.98  358 

1.01  642 

9.99  800 

30 

31 

8.98  288 

131 

8.98  490 

132 

1.01  510 

9.99  798 

29 

2.20 

32 

8.98  419 

131 

8.98  622 

1.01  378 

9.99  797 

28 

131 

2.18 

33 

8.98  549 

130 

8.98  753 

131 

1.01  247 

9.99  796 

27 

130 

2.17 

2.15 

2.13 

2.12 

34 

8.98  679 

129 

8.98  884 

131 

1.01   116 

9.99  795 

26 

128 
127 
126 

35 

8.98  808 

8.99  015 

1.00  985 

9.99  793 

25 

36 

8.98  937 

129 

8.99  145 

130 

1.00  855 

9.99  792 

24 

12s 

2.08 

37 

8.99  066 

129 

8.99  275 

1.00  725 

9.99  791 

23 

124 

2.07 

38 

8.99  194 

8.99  405 

1.00  595 

9.99  790 

?.?. 

123 

2.0s 

39 

8.99  322 

128 
128 

8.99  534 

128 

1.00  466 

9.99  788 

21 

121 
120 

2.03 
2.02 
2.00 

40 

8.99  450 

8.99  662 

1.00  338 

9.99  787 

20 

41 

8.99  577 

127 

8.99  791 

128 

1.00  209 

9.99  786 

19 

42 

8.99  704 

127 

8.99  919 

1.00  081 

9.99  785 

18 

43 

8.99  830 

126 
126 

9.00  046 

128 

0.99  954 

9.99  783 

17 

44 

8.99  956 

9.00  174 

127 

126 
126 
126 

0.99  826 

9.99  782 

16 

45 

9.00  082 

9.00  301 

0.99  699 

9.99  781 

15 

46 

9.00  207 

12s 

9.00  427 

0.99  573 

9.99  780 

14 

47 

9.00  332 

I2S 

9.00  553 

0.99  447 

9.99  778 

13 

48 

9.00  456 

124 

9.00  679 

0.99  321 

9.99  777 

12 

49 

9.00  581 

123 

9.00  805 

125 

0.99  195 

9.99  776 

11 

50 

9.00  704 

9.00  930 

0.99  070 

9.99  775 

10 

51 

9.00  828 

124 

9.01  055 

125 

0.98  945 

9.99  773 

9 

52 

9.00  951 

123 

9.01   179 

124 

0.98  821 

9.99  772 

8 

53 

9.01  074 

123 

9.01  303 

124 

0.98  697 

9.99  771 

7 

54 

9.01  196 

122 

9.01  427 

124 
123 

0.98  573 

9.99  769 

6 

55 

9.01  318 

9.01  550 

0.98  450 

9.99  768 

5 

56 

9.01  440 

122 

9.01  673 

123 

0.98  327 

9.99  767 

4 

57 

9.01  561 

9.01  796 

123 

0.98  204 

9.99  765 

3 

58 

9.01  682 

9.01  918 

0.98  082 

9.99  764 

2 

59 
60 

9.01  803 

120 

9.02  040 

122 

0.97  960 

9.99  763 

1 

9.01  923 

9.02  162 

0.97  838 

9.99  761 

0 

log  COS 

d. 

log  cot 

C.d. 

log  tan 

log  sin 

1 

Prop.  Pts. 

*I74° 

264° 

*354° 

84° 

34 


Logarithms  of  thk  Trigonometric  Functions 


(y 


^96° 


186° 


*276'" 


log  sin   d. 


log  tan 


c.  d,  log  cot 


log  cos 


Prop.  Pts. 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 
160 


9.01  923 

9.02  043 
9.02  163 
9.02  283 
9.02  402 
9.02  520 
9.02  639 
9.02  757 
9.02  874 
9.02  992 


9.03  109 
9.03  226 
9.03  342 
9.03  458 
9.03  574 


9.03  690 
9.03  805 

9.03  920 
9.0^  034 

9.04  149 


9.04  262 
9.04  376 
9.04  490 
9.04  603 
9.04  715 


9.04  828 

9.04  940 

9.05  052 
9.05  164 
9.05  275 


9.05  386 

9.05  497 

9.05  607 

9.05  717 

9.05  827 


9.05  937 

9.06  046 
9.06  155 
9.06  264 
9.06  372 


9.06 
9.06 
9.06 
9.06 
9.06 


481 
589 
696 
804 
911 


9.07 
9.07 
9.07 
9.07 
9.07 


9.07 
9.07 
9.07 
9.07 
9.07 


018 
124 
231 
337 
442 
548 
653 
758 
863 
968 


9.08 
9.08 
9.08 
9.08 
9.08 
9^08" 


072 
176 
280 
383 
486 
589 
log  cos 


120 
120 
120 

119 
118 

119 
118 
117 
118 
117 

117 
116 
116 

n6 
116 


114 
"S 
113 

114 
114 
113 
112 
113 

112 
112 
112 
III 
III 

III 
no 
no 
no 
no 

109 
109 
109 
108 
109 

108 
107 
108 
107 
107 

106 
107 
io6 
105 
106 

loS 
los 


104 
104 
103 
103 
103 

"dT 


^173" 


263° 


9.02  162 
9.02  283 
9.02  404 
9.02  525 
9.02  645 


9.02  766 

9.02  885 

9.03  005 
9.03  124 
9.03  242 


9.03  361 
9.03  479 
9.03  597 
9.03  714 
9.03  832 


9.03  948 

9.04  065 
9.04  181 
9.04  297 
9.04  413 


9.04  528 
9.04  643 
9.04  758 
9.04  873 
9.04  987 


9.05  101 
9.05  214 
9.05  328 
9.05  441 
9.05  553 


9.05  666 
9.05  778 

9.05  890 

9.06  002 
9.06  113 


9.06  224 
9.06  335 
9.06  445 
9.06  556 
9.06  666 


9.06  775 
9.06  885 

9.06  994 

9.07  103 
9.07  211 


9.07  320 

9.07  428 

9.07  536 

9.07  643 

9.07  751 


9.07  858 

9.07  964 

9.08  071 
9.08  177 
9.08  283 


9.08  389 
9.08  495 
9.08  600 
9.08  70S 
9.08  810 
9.08  914 

log  cot 


^353" 


121 
121 
121 
120 

121 

119 
120 
119 
118 
119 

118 
118 

n7 
118 
116 

117 
116 
116 
116 
IIS 

IIS 
IIS 
IIS 
114 
114 

113 
114 
"3 
112 
113 

112 
112 
112 
in 
in 

in 
no 
in 
no 
109 

no 
109 
109 
108 
109 

108 
108 
107 
108 
107 

106 
107 
106 
106 
106 

106 
105 
los 
105 
104 

cTdT 


0.97  838 
0.97  717 
0.97  596 
0.97  475 
0.97  355 


9.99  761 
9.99  760 
9.99  759 
9.99  757 
9.99  756 


0.97  234 
0.97  115 
0.96  995 
0.96  876 
0.96  75S 


9.99  755 
9.99  753 
9.99  752 
9.99  751 
9.99  749 


0.96  639 
0.96  521 
0.96  403 
0.96  286 
0.96  168 


9.99  748 
9.99  747 
9.99  745 
9.99  744 
9.99  742 


0.96  052 
0.95  935 
0.95  819 
0.95  703 
0.95  587 


0.95  472 
0.95  357 
0.95  242 
0.95  127 
0.95  013 


0.94  899 
0.94  786 
0.94  672 
0.94  559 
0.94  447 


0.94  334 
0.94  222 
0.94  110 
0.93  998 
0.93  887 


0.93  776 
0.93  665 
0.93  555 
0.93  444 
0.93  334 


0.93  225 
0.93  115 
0.93  006 
0.92  897 
0.92  789 


0.92  680 
0.92  572 
0.92  464 
0.92  357 
0.92  249 


0.92  142 
0.92  036 
0.91  929 
0.91  823 
0.91  717 


0.91  611 

0.91  505 

0.91  400 

0.91  295 

0.91  190 

0.91  086 

log  tan 

35 


9.99  741 
9.99  740 
9.99  738 
9.99  737 
9.99  736 


9.99  734 
9.99  733 
9.99  731 
9.99  730 
9.99  728 


9.99  727 
9.99  726 
9.99  724 
9.99  723 
9.99  721 


9.99  720 
9.99  718 
9.99  717 
9.99  716 
9.99  714 


9.99  713 
9.99  711 
9.99  710 
9.99  708 
9.99  707 


9.99  705 
9.99  704 
9.99  702 
9.99  701 
9.99  699 


9.99  698 
9.99  696 
9.99  695 
9.99  693 
9.99  692 


9.99  690 
9.99  689 
9.99  687 
9.99  686 
9.99  684 


9.99  683 
9.99  681 
9.99  680 
9.99  678 
9.99  677 
9.99  675 


log  sin  I  ' 


// 

121 

120 

6 

12. 1 

12.0 

7 

14.1 

14.0 

8 

ib.i 

16.0 

0 

18.2 

18.0 

10 

20.2 

20.0 

20 

40.3 

40.0 

,^0 

60.  s 

60.0 

40 

80.7 

80.0 

50 

100.8 

lOO.O  I 

// 

118 

117 

6 

11.8 

II. 7 

7 

13.8 

13.6 

8 

15.7 

15.6 

9 

17.7 

17.6 

10 

19.7 

19.S 

20 

39-3 

39.0 

30 

S9.0 

SS.,'! 

40 

78.7 

78.0 

50 

9«.3 

97.S 

" 

115 

114 

6 

II. S 

11.4 

7 

13.4 

13.3 

8 

1 5.3 

IS.2 

9 

17.2 

I7.I 

10 

19.2 

19.0 

20 

38.3 

38.0 

,30 

57.  s 

S7.0 

40 

76.7 

76.0 

5° 

95.« 

9S.O 

" 

112 

111 

6 

II. 2 

II. I 

7 
8 
9 
10 

13-1 
14.9 
16.8 
18.7 

13.0 

14.8 
16.6 
18..S 

20 
30 
40 
SO 

37.3 
s6.o 
74.7 
93-3 

37.0 
5S.S 
74.0 
92.S 

6 

109 

10.9 

108 

10.8 

7 

12.7 

12.6 

8 

14.  s 

14.4 

9 

16.4 

16.2 

10 

18.2 

18.0 

20     36.3 

36.0 

30     54.  S 

S4.0 

40     72.7 

72.0 

50 

90.8 

90.0 

10.6 
12.4 

I4.I 
15.9 
17.7 
35.3 
53.0 
70.7 
88.3 


lo.S 
12.2 
14.0 
1S.8 


11.9 
13.9 
15.9 

17.8 
19.8 
39-7 
59.5 
79.3 


116 

II. 6 
13. S 
iS-S 
17.4 
19-3 
38.7 
s8.o 
77.3 
96.7 


113 

II. 3 
13.2 
iS-i 
17.0 
18.8 
37.7 
S6.S 
7S.3 
94.2 


110 

II.O 

12.8 
14.7 
16.S 
18.3 
36.7 
SS.o 
73.3 
91.7 


10.7 

1 2.5 

14.3 

16.0 

17.8 

3S.7 
53-5 
71.3 
89.2 


106  105  104 


10.4 
12. 1 
13.9 
1S.6 


17.S  17.3 
3S.O  34.7 

S2.S   S2.0 

70.0  69.3 

87.S  1 86.7 


Prop.  Pts. 


Logarithms  of  the  Trigonometric  Functions 


*97 


187'^        *277" 


log  sin 


d. 


log  tan 


c.  (1. 


log  cot 


log  cos 


Prop.  Pts. 


9.08  589 
9.08  692 
9.08  795 
9.08  897 
9.08  999 


9.09  101 
9.09  202 
9.09  30+ 
9.09  405 
9.09  506 


9.09  6U0 
9.09  707 
9.09  807 

9.09  907 

9.10  006 


9.10  lOo 
9.10  205 
9.10  30+ 
9.10  402 
9.10  501 


20 

21 
22 
23 

25 
26 
27 
28 
29 


9.10  599 
9.10  697 
9.10  795 
9.10  893 
9.10  990 


9.11 
9.11 
9.11 
9.11 
9.11 


087 
18+ 
281 
377 
474 


30 

31 
32 
33 

ii 

35 
36 
37 
38 

40 

41 

42 
+3 
4+ 


9.11 
9.11 
9.11 
9.11 
9.11 


570 
666 
761 

857 
952 


9.12  0+7 
9.12  142 
9.12  236 
9.12  331 
9.12  425 


9.12  519 
9.12  612 
9.12  706 
9.12  799 
9.12  892 


9.12  985 

9.13  078 
9.13  171 
9.13  263 
9.13  355 


50 

51 

52 
53 

il 
55 
56 
57 
58 
59 


9.13 
9.13 
9.13 
9.13 
9.13 


4+7 
539 
630 
722 
813 


9.13  90+ 
9.13  99+ 
9.1+  085 
9.1+  175 
9.1+  266 


60    9.1+  356 


103 
103 
102 


102 

lOI 
lOI 

100 

lOI 

100 
100 
99 
100 

99 
99 


97 
97 

97 
97 
96 
97 
96 

96 
OS 
96 
95 
9S 

95 
94 
95 
94 
94 

93 
94 
93 
93 
93 

93 
93 
92 
92 
92 

92 
91 
92 
91 
91 

90 
91 
90 
91 
90 


9.08 
9.09 
9.09 
9.09 
9.09 


914 
019 
123 

227 
330 


9.09 
9.09 
9.09 
9.09 
9.09 


434 
537 
640 
742 

84i 


9.09 
9.10 
9.10 
9.10 
9.10 


947 
0+9 
150 
252 
353 


9.10 
9.10 
9.10 
9.10 
9.10 


454 
555 
656 
756 
856 


9.10 
9.11 
9.11 
9.11 
9.11 


956 
056 
155 
254 
353 


9.11 
9.11 
9.11 
9.11 
9.11 


452 
551 
649 
747 

845 


9.11 
9.12 
9.12 
9.12 
9.12 


943 
040 
138 
235 
332 


9.12 
9.12 
9.12 
9.12 
9.12 


428 
525 
621 
717 
813 


9.12 
9.13 
9.13 
9.13 
9.13 


909 
00+ 
099 
19+ 
289 


9.13 
9.13 
9.13 
9.13 
9.13 


38+ 
478 
573 
667 
761 


9.13 
9.13 
9.14 
9.14 
9.14 


85+ 
9+8 
0+1 
13+ 
227 


9.1+ 
9.14 
9.14 
9.14 
9.14 


320 
412 
504 
597 


9.14  780 


10s 
104 
104 
103 
104 

103 

103 
102 
103 
102 

102 

lOI 

102 

lOI 
lOI 


lOI 

100 
100 
100 

100 
99 
99 
99 
99 


98 


97 
98 
97 
97 
96 

97 
96 
96 
96 
96 

95 
95 
95 
95 
95 

94 
95 
94 
94 
93 

94 
93 
93 
93 
93 

92 
92 
93 
91 
92 


0.91  086 

0.90  981 
0.90  877 
0.90  773 
0.90  670 


9.99 
9.99 
9.99 
9.99 
9.99 


675 
674 
672 
670 
669 


0.90  566 
0.90  463 
0.90  360 
0.90  258 
0.90  155 


9.99 
9.99 
9.99 
9.99 
9.99 


667 
666 
664 
663 
661 


0.90  053 
0.89  951 
0.89  850 
0.89  748 
0.89  647 


9.99 
9.99 
9.99 
9.99 
9.99 


0.89  546 

0.89  445 

0.89  344 

0.89  244 

0.89  144 


9.99 
9.99 
9.99 
9.99 
9.99 


659 
658 
656 
655 
653 
651 
650 
648 
647 
645 


0.89  044 
0.88  944 
0.88  8+5 
0.88  746 
0.88  647 


9.99 
9.99 
9.99 
9.99 
9.99 


643 
642 
640 
638 
637 


0.88  548 

0.88  449 

0.88  351 

0.88  253 

0.88  155 


9.99 
9.99 
9.99 
9.99 
9.99 


635 
633 
632 
630 
629 


0.88  057 
0.87  960 
0.87  862 
0.87  765 
0  87  668 


9.99 
9.99 
9.99 
9.99 
9.99 


627 
625 
624 
622 
620 


0.87 
0.87 


572 
475 


0.87  379 
0.87  283 
0.87  187 


9  99 
9.99 
9.99 
9.99 
9.99 


618 
617 
615 
613 
612 


0.87  091 
0.86  996 


0.86 
0.86 
0.86 


901 
806 
711 


9.99 
9.99 
9.99 
9.99 
9.99 


610 
608 
607 
605 
603 


0.86  616 
0.86  522 
0.86  427 
0.86  333 
0.86  239 


9.99 
9.99 
9.99 
9.99 
9.99 


601 
600 
598 
596 
595 


0.86  146 
0.86  052 


0.85 
0.85 
0.85 


0.85 
0.85 
0.85 
0.85  403 
0.85  312 


959 
866 
773 
680 
588 
496 


9.99 
9.99 
9.99 
9.99 
9.99 
9.99 
9.99 
9.99 
9.99 
9.99 


593 
591 

589 
588 
586 
584 
582 
581 
579 
577 


0.85   220 


9.99  575 


"  106  104  108 


lo.S 
12.3 
14.0 
1S.8 
17.S 
3S.O 
S2.5 
70.0 
87.5 


10.4 
12. 1 
13.9 
15.6 
17.3 
34-7 
S2.0 
69.3 
86.7 


10.3 
12.0 
13.7 
IS.4 
17.2 
34.3 
Si.S 

68.7 
85.8 


102  101  100 


10.2 

II.9 
13.6 
15-3 
17.0 
34-0 


10. 1 
11.8 
13.5 
15.2 
16.8 
33.7 


51-050.5 
68.067.3 
8s.o!84.2 


lO.O 

II. 7 
13.3 
15.0 
16.7 
33-3 
50.0 
66.7 
83.3 


"  99    98    97 


9.9 
11.6 
13.2 
14.8 
16.5 
33.0 
49.5 
66.0 


9.8 
11.4 
13.1 

14.7 
16.3 
32.7 
49.0 
65.3 


82.si8i.7 


9.7 
11.3 
12.9 
14.6 
16.2 
32.3 
48. 5 
64.7 


9.6 

II. 2 
12.8 
14.4 
16.0 
32.0 


95    94 


9-5 
II. I 
12.7 
14.2 
1S.8 
31.7 


48.0 1 47. 5 
64.0  63.3 
80.0 179. 2 


9.4 

II.O 

12.S 
14.1 
15-7 
31.3 
47.0 
62.7 
78.3 


"  93    92    91 


9.3 
10.9 

12.4 
14.0 
15-5 
3I.O 
46.5 
62.0 
77-5 


9.2 
10.7 
12.3 
13.8 
15-3 
30-7 
46.0 
61.3 
76.7 


" 

90 

2 

6 

0.0 

0.2 

7 

10. 5 

0.2 

8 

12.0 

0.3 

9 

13.S 

0.3 

10 

1.5.0 

0.3 

20 

30.0 

0.7 

30 

4S.O 

I.O 

40 

60.0 

1.3 

SO 

75.0 

1.7 

9.1 

10.6 
12. 1 
13.6 
15.2 
30.3 
45.5 
60.7 
75.8 


lof?  cos 


^172° 


d.      log  cot    c. d 


262° 


^352" 


log  tan      log  sin 

36 


Prop.  Pts. 


Logarithms  of  the  Trigonometric  Functions 


8° 


*278" 


log  sill      d. 


log  tan    c.  d. 


log  cot 


log  cos 


Prop.  Pts. 


9.14  356 
9.14  445 
9.14  535 
9.14  624 
9.14  714 


20 

21 
22 
23 
24 


60 


9.14 
9.14 
9.14 
9.15 
9.15 


803 
891 
980 
069 
157 


9.15 
9.15 
9.15 
9.15 
9.15 


245 
333 
421 
508 
596 


9.15 
9.15 
9.15 
9.15 
9.16 


683 
770 
857 
944 
030 


9.16 
9.16 
9.16 
9.16 
9.16 


9.16 
9.16 
9.16 
9.16 
9.16 


9.16 
9.17 
9.17 
9.17 
9.17 


116 
203 
289 
374 
460 
545 
631 
716 
801 
886 
970 
055 
139 
223 
307 


9.17 
9.17 
9.17 
9.17 
9.17 


9.17 
9.17 
9.17 
9.18 
9.18 


9.18 
9.18 
9.18 
9.18 
9.18 


9.18 
9.18 
9.18 
9.18 
9.18 


9.19 
9.19 
9.19 
9.19 
9.19 


391 
474 
558 
641 

724 

807 
890 
973 
055 
137 
220 
302 
383 
465 
547 
628 
709 
790 
871 
952 
033 
113 
193 
273 
353 


9.19  433 


87 
88 

87 

87 
87 
87 

86 
86 

87 
86 
8S 
86 
85 
86 
8S 
8s 
85 
84 

8S 
84 
84 
84 
84 

83 
84 
83 
83 
83 

83 
83 
82 
82 
83 

82 
81 
82 
82 


9.14 
9.14 
9.14 
9.15 
9.15 


780 
872 
963 
054 
145 


9.15 
9.15 
9.15 
9.15 
9.15 


236 
327 

417 
508 
598 


9.15 
9.15 
9.15 
9.15 
9.16 


688 
777 
867 
956 
046 


9.16  135 
9.16  224 
9.16  312 
9.16  401 
9.16  489 


9.16  577 
9.16  665 
9.16  753 
9.16  841 
9.16  928 


9.17  016 
9.17  103 
9.17  190 
9.17  277 
9.17  .363 


9.17  450 
9.17  536 
9.17  622 
9.17  708 
9.17  794 


9.17  880 

9.17  965 

9.18  051 
9.18  136 
9.18  221 
9.18  306 
9.18  391 
9.18  475 
9.18  560 
9.18  644 


9.18  728 
9.18  812 
9.18  896 

9.18  979 

9.19  063 


9.19  146 
9.19  229 
9.19  312 
9.19  395 
9.19  478 


9.19  561 
9.19  643 
9.19  725 
9.19  807 
9.19  889 


9.19  971 


92 
91 
91 
91 

91 

91 
90 
91 
90 
90 

89 
90 


89 


87 
88 

87 
87 
87 
86 
87 

86 
86 
86 
86 
86 

8S 
86 
8S 
8S 
8S 

8S 
84 
85 
84 
84 

84 
84 
83 
84 
83 

83 
83 
83 
83 
83 

82 
82 
82 
82 


0.85  220 
0.85  128 
0.85  037 
0.84  946 
0.84  855 


9.99  575 
9.99  574 
9.99  572 
9.99  570 
9.99  568 


0.84  764 

0.84  673 

0.84  583 

0.84  492 

0.84  402 


0.84  312 
0.84  223 
0.84  133 
0.84  044 
0.83  954 


0.83  865 
0.83  776 


0.83 
0.83 
0.83 


688 
599 
511 


0.83  423 
0.83  335 
0.83  247 
0.83  159 
0.83  072 


0.82  984 
0.82  897 
0.82  810 
0.82  723 
0.82  637 


0.82  550 
0.82  464 
0.82  378 
0.82  292 
0.82  206 


0.82  120 
0.82  035 
0  81  949 
0.81  864 
0.81  779 


0.81  694 

0.81  609 

0.81  525 

0.81  440 

0.81  356 


0.81  272 
0.81  188 
0.81  104 
0.81  021 
0.80  937 


0.80  854 
0.80  771 
0.80  688 
0.80  605 
0.80  522 


0.80  439 
0.80  357 
0.80  275 
0.80  193 
0.80  111 


9.99  566 

9.99  565 

9.99  563 

9.99  561 

9.99  559 


9.99  557 
9.99  556 
9.99  554 
9.99  552 
9.99  550 


9.99  548 
9.99  546 
9.99  545 
9.99  543 
9.99  541 


9.99  539 

9.99  537 

9.99  535 

9.99  533 

9.99  532 


9.99  530 
9.99  528 
9.99  526 
9.99  524 
9.99  522 


9.99 
9.99 
9.99 
9.99 
9.99 


520 
518 
517 
515 
513 


9.99 
9.99 
9.99 
9.99 
9.99 


511 
509 
507 
505 
503 


9.99 
9.99 
9.99 
9.99 
9.99 


501 
499 
497 
495 
494 


9.99 
9.99 
9.99 
9.99 
9.99 


492 
490 
488 
486 
484 


9.99 
9.99 
9.99 
9.99 
9.99 


482 
480 
478 
476 
474 


9.99 
9.99 
9.99 
9.99 
9.99 


472 

470 
468 
466 
464 


0.80  029 


9.99  462 


92  I  91    90 

6    9.2    9.1    9.0 

710.7,10.6110.5 

8112.3J12.1  12.0 

9li3.8!i3.6  13.5 

1015.3  15.2  15.0 

20  30.7'30.3'30.o 

30,46.0  45.5  45-0 

40161.3,60.7:60.0 

So!76.7l7S.8!7S.o 


89  I  88    87 

6  8.9    8.8    8.7 

7  I0.4  10.3  10.2 

8  ii.9|ii.7jii.6 
9'i3-4  I3.2;i3.i 

10  14.8  14.714.3 
20129.7I29.3J29.0 
30  44.5,44-0  43  5 
40 1 59.3  58.7  j  58.0 
S0I74.2I73.3I72.S 


86    85  i  84 


8.6 
10.0 


8.5    8.4 

9.9    9.8 

11.5  11.3I11.2 

12.9:12.8(12.6 

|l4.3'l4.2|i4-0 
20J28.7  28.3J28.0 
3o:43.o|42.5'42.o 
40I57-3  56.7:56.0 
So|7i.7i 70.8' 70.0 


83    82    81 


8.3 
9-7 
II. I 

,  I2.i 

13.8 

27.7 
30,41.5:41-0 
40,55.3  54-7 
50169.2168.3 


9.5 
10.8 
12.2 
I3-S 
27.0 
40.5 
54-0 
67.5 


"  80     2      1 


0.1 
0.1 
0.1 
0.2 
0.2 
0.3 
0.5 
0.7 
0.8 


log  COS      d. 


261° 


log  cot    c.  d. !  log  tan 


log  sin 


'351- 


81° 

37 


Prop.  Pts. 


LOGAKITHMS    OF    THE    TRIGONOMETRIC    FUNCTIONS 


9^ 


•^99 


*279° 


log  sin 


d. 


log  tan    c.  d 


log  cot      log  cos 


Prop.  Pts. 


9.19 
9.19 
9.19 
9.19 
9.19 


433 
513 
592 
672 
751 


9.19 
9.19 
9.19 
9.20 
9.20 


830 
909 
988 
067 
145 


10 

11 
12 
13 
14 
15 
16 
17 
18 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 

il 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 


9.20 
9.20 
9.20 
9.20 
9.20 


223 
302 
380 

458 
535 


50 

51 

52 
53 
54 
55 
56 
57 
58 

11 
60 


9.20 
9.20 
9.20 
9.20 
9.20 


613 
691 

768 
845 
922 


9.20 
9.21 
9.21 
9.21 
9.21 


999 
076 
153 
229 
306 


9.21 
9.21 
9.21 
9.21 
9.21 


382 
458 
534 
610 
685 


9.21 
9.21 
9.21 
9.21 
9.22 


761 
836 
912 
987 
062 


9.22 
9.22 
9.22 
9.22 
9.22 


9.22 
9.22 
9.22 
9.22 
9.22 


509 
583 
657 
731 
805 


9.22 
9.22 
9.23 
9.23 
9.23 


878 
952 
025 
098 
171 


9.23 
9.23 
9.23 
9.23 
9.23 


244 
317 
390 
462 
535 


9.23 
9.23 
9.23 
9.23 
9.23 


137 
211 
286 
361 
435 


607 
679 
752 
823 
895 


9.23  967 
log  COS 


8o 
79 
8o 

79 
79 

79 
79 
79 
78 
78 

79 
78 
78 
77 
78 

78 
77 
77 
77 
77 

77 
77 
76 
77 
76 

76 
76 
76 
75 
76 

7S 
76 
75 
75 
75 

74 
75 
75 
74 
74 

74 
74 
74 
74 
73 

74 
73 
73 
73 
73 

73 
73 
72 
73 
72 

72 
73 
71 
72 
72 

"dT 


9.19 
9.20 
9.20 
9.20 
9.20 


971 
053 
134 
216 
297 


9.20 
9.20 
9.20 
9.20 
9.20 


378 
459 
540 
621 
701 


9.20 
9.20 
9.20 
9.21 
9.21 


782 
862 
942 
022 
102 


9.21 
9.21 
9.21 
9.21 
9.21 


182 
261 
341 
420 
499 


9.21 
9.21 
9.21 
9.21 
9.21 


578 
657 
736 
814 
893 


9.21 
9.22 
9.22 
9.22 
9.22 


971 
049 
127 
205 
283 


9.22 
9.22 
9.22 
9.22 
9.22 


361 
438 
516 
593 
670 


9.22 
9.22 
9.22 
9.22 
9.23 


747 
824 
901 
977 
054 


9.23 
9.23 
9.23 
9.23 
9.23 


130 
206 
283 
359 
435 


9.23 
9.23 
9.23 
9.23 
9.23 


510 
586 
661 
737 
812 


9-23 
9.23 
9.24 
9.24 
9.24 


887 
962 
037 
112 
186 


9.24 
9.24 
9.24 
9.24 
9.24 


261 
335 
410 

484 

558 


82 
81 
82 
81 
81 

81 
81 
81 
80 
81 

80 
80 
80 
80 
80 

79 
80 
79 
79 
79 

79 
79 
78 
79 
78 

78 
78 
78 
78 
78 

77 
78 
77 
77 
77 

77 
77 
76 
77 
76 

76 
77 
76 
76 
75 
76 
75 
76 
75 
75 

75 
75 
75 
74 
75 

74 
75 
74 
74 
74 


9.24  632 
log  cot    C.  d 


0.80  029 
0.79  947 
0.79  866 
0.79  784 
0.79  703 


9.99  462 
9.99  460 
9.99  458 
9.99  456 
9.99  454 


0.79  622 
0.79  541 
0.79  460 
0.79  379 
0.79  299 


9.99  452 
9.99  450 
9.99  448 
9.99  446 
9.99  444 


0.79  218 

0.79  138 

0.79  058 

0.78  978 

0.78  898 


9.99  442 
9.99  440 
9.99  438 
9.99  436 
9.99  434 


0.78  818 
0.78  739 
0.78  659 
0.78  580 
0.78  501 


9.99  432 
9.99  429 
9.99  427 
9.99  425 
9.99  423 


0.7S  422 
0.78  343 
0.78  264 
0.78  186 
0.78  107 


9.99  421 
9.99  419 
9.99  417 
9.99  415 
9.99  413 


0.78  029 
0.77  951 
0.77  873 
0.77  795 
0.77  717 


9.99  411 
9.99  409 
9.99  407 
9.99  404 
9.99  402 


0.77  639 
0.77  562 
0.77  484 
0.77  407 
0.77  330 


9.99  400 
9.99  398 
9.99  396 
9.99  394 
9.99  392 


0.77  253 
0.77  176 
0.77  099 
0.77  023 
0.76  946 


9.99  390 
9.99  388 
9.99  385 
9.99  383 
9.99  381 


0.76  870 
0.76  794 
0.76  717 
0.76  641 
0.76  565 


9.99  379 
9.99  377 
9.99  375 
9.99  372 
9.99  370 


0.76  490 

0.76  414 

0.76  339 

0.76  263 

0.76  188 


9.99  368 
9.99  366 
9.99  364 
9.99  362 
9.99  359 


0.76  113 
0.76  038 


0.75 
0.75 
0.75 


963 
888 
814 


9.99  357 
9.99  355 
9.99  353 
9.99  351 
9.99  348 


^70" 


260=^ 


*35o" 


0.75  739 

0.75  665 

0.75  590 

0.75  516 

0.75  442 

0.75  368 

log  tan 

"8(F" 


9.99  346 
9.99  344 
9.99  342 
9.99  340 
9.99  337 
9.99  335 

log  sin 


80 

79 

8.0 

7.9 

9-3 

92 

10.7 

10. 5 

12.0 

II.Q 

i,^.,S 

13.2 

26.7 

26.3 

40.0 

39-5 

53.3 

.52.7 

66.7 

65.8 

77 

76 

7-7 

7.6 

9.0 

8.9 

10.3 

10. 1 

11.6 

11.4 

12.8 

12.7 

25.7 

25.3 

38.  S 

38.0 

SI.3 

.50.7 

64.2 

bi.3 

78 

7.8 
9.1 
10.4 
11.7 
13.0 
26.0 
39.0 
52.0 
65.0 


75 

7.5 
8.8 

lO.O 

"•3 
12.5 
25.0 
37.5 
So.o 
62.S 


74 


7.4 
8.6 
9.9 
II. I 
10I12.3 
20J24.7 
30,37.0 
4049-3 
50,61.7 


78 

7.3 
8.5 
9-7 

II.O 

12.2 
24-3 
36.S 
48.7 
60.8  60.0 


72 

7.2 
8.4 
9.6 
10.8 
12.0 
24.0 
36.0 
48.0 


71 

8 

7.1 

o.,i 

8.3 

0.4 

9S 

0.4 

10.7 

0.5 

11.8 

o.S 

23.7 

I.O 

35.5 

1-5 

47.3 

2.0 

59-2 

2.5 

Prop.  Pts. 


S8 


Logarithms  of  the  Trigoxometkic  Fuxctions 


10 


I90' 


*2So^ 


0 

1 

2 
3 

5 

6 

7 

8 

9^ 

10 

11 

12 

13 

2+ 

15 

16 

1; 

IS 

19 


20 
21 
22 
23 
24 
"25" 
26 
27 
28 
29 
30 
31 
32 
33 

it 
35 
36 

37 
38 
39^ 

40 

41 
42 
43 

ii 

45 
46 
47 
48 
49 


50 

51 
52 
53 

ii 
55 
56 
57 
58 
_59 
60 


Io£r  sin       (1. 


9.23 
9.24 
9.24 
9.24 
9.24 


967 
039 
110 
181 
253 


9.24 
9.24 
9.24 
9.24 
9.24 


324 
395 
466 
536 
607 


9.24 
9.24 
9.24 
9.24 
9.24 


677 

748 
818 
888 
958 


9.25 
9.25 
9.25 
9.25 
9.25 


028 
098 
168 
237 
307 


9.25 
9.25 
9.25 
9.25 
9.25 


9.25 
9.25 
9.25 
9.25 
9.25 


376  ! 

445 

514 

583 

652 

721 

790 

858  I 

927 

995 


9.26 
9.26 
9.26 
9.26 
9.26 


063 
131 
199 
267 
335 


9.26 
9.26 
9.26 
9.26 
9.26 


403 
470 
538 
605 
672 


9.26 
9.26 
9.26 
9.26 
9.27 


739 
806 
873 
940 
007 


9.27 
9.27 
9.27 
9.27 
9.27 


073 
140 
206 
273 
339 


9.27 
9.27 
9.27 
9.27 
9.27 


405 
471 
537 
602 
668 


9.27 
9.27 
9.27 
9.27 
9.27 


734 
799 
864 
930 
995 


9.28  060 


70 
70 
69 
70 
69 

69 
69 
69 
69 
69 

69 
68 
69 
68 
68 

68 
68 
68 
68 
68 

67 
68 
67 
67 
67 

67 
67 
67 
67 
66 

67 
66 
67 
66 
66 

66 
66 
65 
66 
66 

6S 
65 
66 
63 
65 


loe  COS      d. 


^169= 


259^^ 


log  tan    f.  (1.     log  cot 


9.24 
9.24 
9.24 
9.24 
9.24 


632 
706 
779 
853 
926 


9.25 
9.25 
9.25 
9.25 
9.25 


000 
073 
146 
219 
292 


9.25 
9.25 
9.25 
9.25 
9.25 
9.25 
9.25 
9.25 
9.25 
9.26 


365 
437 
510 
582 
655 


727 
799 
871 
943 
015 


9.26 
9.26 
9.26 
9.26 
9.26 


086 
158 
229 
301 
372 


9.26 
9.26 
9.26 
9.26 
9.26 


443 
514 
585 
655 
726 


9.26 
9.26 
9.26 
9.27 
9.27 


797 
867 
937 
008 
078 


9.27 
9.27 
9.27 
9.27 
9.27 


148 
218 
288 
357 
427 


9.27 
9.27 
9.27 
9.27 
9.27 


496 
566 
635 
704 
773 


9.27 
9.27 
9.27 
9.28 
9.28 
9.28 
9.28 
9.28 
9.28 
9.28 


842 
911 
980 
049 
117 


186 
254 
323 
391 
459 


9.28 
9.28 
9.28 
9.28 
9.28 


527 
595 
662 
730 
798 


9.2s  865 


74 
73 
74 
73 
74 

73 
73 
73 
73 
73 

72 
73 
72 
73 
72 

72 
72 
72 
72 

71 

72 
71 
72 
71 
71 

7» 
71 
70 
71 
71 


69 
70 
69 

70 
69 
69 
69 
69 

69 
69 
69 
68 
69 

68 
69 
68 
68 
68 

68 
67 
68 

68 
67 


0.75  36S 
0.75  294 
0.75  221 
0.75  147 
0.75  074 


0.75  000 

0.74  927 

0.74  854 

0.74  781 

0.74  708 


0.74  635 
0.74  563 
0.74  490 
0.74  418 
0.74  345 


0.74 
0.74 
0.74 
0.74 
0.73 


273 
201 
129 
057 
985 


0.73 
0.73 
0.73 
0.73 
0.73 


914 
842 
771 
699 
628 


0.73 
0.73 
0.73 
0.73 
0.73 


557 
486 
415 
345 
274 


0.73 
0.73 
0.73 
0.72 
0.72 


203 
133 
063 
992 
922 


0.72  852 

0.72  782 

0.72  712 

0.72  643 

0.72  573 


0.72 
0.72 
0.72 
0.72 
0.72 


504 
434 
365 
296 
227 


0.72 
0.72 
0.72 
0.71 
0.71 


1.58 
089 
020 
951 
883 


0.71 
0.71 
0.71 
0.71 
0.71 


814 
746 
677 
609 
541 


0.71 
0.71 
0.71 
0.71 
0.71 


473 
405 
338 
270 
202 


0.71   135 


log  cot    c.  d.     log  tau 


log  cos 


9.99 
9.99 
9.99 
9.99 
9.99 


333 
331 
328 
326 


9.99 
9.99 
9.99 
9.99 
9.99 


324 
322 
319 
317 
315 


9.99 
9.99 
9.99 
9.99 
9.99 


313 
310 
308 
306 
304 


9.99 
9.99 
9.99 
9.99 
9.99 


301 
299 
297 
294 
292 


9.99 
9.99 
9.99 
9.99 
9.99 


290 
288 
285 
283 
281 


9.99 
9.99 
9.99 
9.99 
9.99 


278 
276 
274 
271 
269 


9.99 
9.99 
9.99 
9.99 
9.99 


267 
264 
262 
260 
257 


9.99  255 
9.99  252 
9.99  250 
9.99  248 
9.99  245 


9.99  243 
9.99  241 
9.99  238 
9.99  236 
9.99  233 


9.99  231 

9.99  229 

9.99  226 

9.99  224 

9.99  221 


9.99  219 
9.99  217 
9.99  214 
9.99  212 
9.99  209 


9.99  207 
9.99  204 
9.99  202 
9.99  200 
9.99  197 


9.99  195 


los  sill 


^^349' 


79^ 

39 


Prop.  Pts. 


6    7.4    7.3    7.2 
r    8.6    8.5    8.4 

8  9.9    9.7    9.6 

9  II. I  ii.o  10.8 
10  12.3  12.2  12.0 
20; 24.7  24.3  24.0 
30  37.0  36.5  36.0 
40  49.3  48.7  48.0 
50:61.7  60.860.0 


71 


7.J 
8.3 
9-S 
10.7 
11.8 
20123.7 
3035.5 
4047-3 
50SS9.2 


6.9 
8.0 
9-2 


70 

7.0 

8.2 

9-3 
10.5,10.4 
11.7:11.5 
23-3,23-0 
35-0  34-5 
46.7:46.0 
S8.3!S7.S 


68    C7  i  66 


6.8 

7.9 

9.1 

1 10.2 

11.3 

i  22.7 

3o'34-0 

4045.3 

50.56.7 


6.7  6.6 

7.8  7.7 

8.9  8.8 
10.0  9.9 

II. 2  II.O 
22.3|22.0 
33.533-0 
44.7  44.0 
55.8.SS.O 


^J  66  I  SI  2 

6  6.5;  0.3    0.2 

7  7.6;  0.4  : 0.2 

8  8.7,0.4:0.3 

9  9.8  0.i  0.3 
10  10.8;  0.5  0.3 
20  21.7  i.o  I  0.7 
3o|32.s|  1.5  '  1.0 
40,43-3 1  2.0  j  1.3 
So!S4-2l  2.5  I  1.7 


Prop.  Pts. 


Logarithms  of  the  TrigoxomeTrio  Functions 


11 


191 


*28l° 


lo?  sin 


(I. 


log  tan 


c.  d. 


log  cot 


log  cos 


Prop.  Pts. 


20 

21 
22 
23 
24 


25 
26 

27 
28 

^ 
30 

31 
32 
33 
34 


45 
46 
47 
48 
49 
50 
51 
52 
53 

ii 

55 
56 

57 
58 

11 
60 


9.28  060 
9.28  125 
9.28  190 
9.28  254 
9.28  319 


9.28 
9.28 
9.28 
9.28 
9.28 


384 
448 
512 
577 
641 


9.28 
9.28 
9.28 
9.28 
9.28 


705 
769 
833 
896 
960 


9.29 
9.29 
9.29 
9.29 
9.29 


024 
087 
150 
214 

277 


9.29 
9.29 
9.29 
9.29 
9.29 


340 
403 
466 
529 
591 


9.29 
9.29 
9.29 
9.29 
9.29 


654 
716 
779 
841 
903 


9.29 
9.30 
9.30 
9.30 
9.30 


966 
028 
090 
151 
213 


9.30 
9.30 
9.30 
9.30 
9.30 


275 
336 
398 
459 
521 


9.30 
9.30 
9.30 
9.30 
9.30 


582 
643 
704 
765 

826 


9.30 
9.30 
9.31 
9.31 
9.31 


8S/ 
947  I 
008  ' 
068  i 
129  i 


9.31 
9.31 
9.31 
9.31 
9.31 


189 
250 
310 
370 
430 


9.31 
9.31 
9.31 
9.31 
9.31 


490 
549 
609 
669 

728 


9.31   78S 


log  cos 


6s 
6s 
64 
6S 
65 

64 
64 
6S 
64 
64 

64 
64 
63 
64 
64 

63 
63 
64 
63 
63 

63 
63 
63 
62 
63 
62 
63 
62 
62 
63 

62 
62 
61 
62 
62 

61 
62 
61 
62 
61 

61 
61 
6t 
61 
61 

60 
6i 
60 
61 
60 

61 
60 
60 
60 
60 

59 
60 
60 
59 
60 


9.28 
9.28 
9.29 
9.29 
9.29 


865 
933 
000 
067 
134 


9.29 
9.29 
9.29 
9.29 
9.29 


201 
268 
335 
402 
468 


9.29 
9.29 
9.29 
9.29 
9.29 


535 
601 
668 
734 
800 


9.29 
9.29 
9.29 
9.30 
9.30 


8bo 
932 
998 
064 
130 


9.30 
9.30 
9.30 
9.30 
9.30 


195 
261 
326 
391 
457 


9.30 
9.30 
9.30 
9.30 
9.30 


522 
587 
652 
717 

782 


9.30 
9.30 
9.30 
9.31 
9.31 


846 
911 
975 
040 
104 


9.31 
9.31 
9.31 
9.31 
9.31 


168 
233 
297 
361 

425 


9.31 
9.31 
9.31 
9.31 
9.31 


489 
552 
616 
679 
743 


9.31 
9.31 
9.31 
9.31 
9.32 


806 
870 
933 
996 
059 


9.32 
9.32 
9.32 
9.32 
9.32 


122 
185 
248 
311 
373 


9.32 
9.32 
9.32 
9.32 
9.32 


436 
498 
561 
623 
685 


9.32  747 


68 

67 
67 
67 
67 

67 
67 
67 

66 
67 

66 
67 
66 
66 
66 

66 
66 
66 
66 
6S 
66 
65 
65 
66 
65 

65 
6S 
65 
65 
64 

6S 
64 
65 
64 
64 

65 
64 
64 
64 
64 

63 
64 
63 
64 
63 

64 
63 
63 
63 
63 

63 
63 
63 
62 
63 

62 
63 
62 
62 
62 


0.71  135 
0.71  067 
0.71  000 
0.70  933 
0.70  866 


9.99  195 

9.99  192 

9.99  190 

9.99  187 

9.99  185 


0.70  799 

0.70  732 

0.70  665 

0.70  598 

0.70  532 


9.99  182 

9.99  180 

9.99  177 

9.99  175 

9.99  172 


0.70  465 

0.70  399 

0.70  332 

0.70  266 

0.70  200 


9.99  170 

9.99  167 

9.99  165 

9.99  162 

9.99  160 


0.70  134 
0.70  068 
0.70  002 
0.69  936 
0.69  870 


9.99  157 

9.99  155 

9.99  152 

9.99  150 

9.99  147 


0.69  805 

0.69  739 

0.69  674 

0.69  609 

0.69  543 


9.99  145 

9.99  142 

9.99  140 

9.99  137 

9.99  135 


0.69  478 

0.69  413 

0.69  348 

0.69  283 

0.69  218 


9.99  132 

9.99  130 

9.99  127 

9.99  124 

9.99  122 


0.69  154 
0.69  089 
0.69  025 
0  68  960 
0.68  896 


9.99  119 

9.99  117 

9.99  114 

9.99  112 

9.99  109 


0.68  832 

0.68  767 

0.68  703 

0.68  639 

0.68  575 


9.99  106 
9.99  104 
9.99  101 
9.99  099 
9.99  096 


0.68  511 

0.68  448 

0.68  384 

0.68  321 

0.68  257 


9.99  093 
9.99  091 
9.99  088 
9.99  086 
9.99  083 


0.68 
0.68 


194 
130 


0.68  067 
0.68  004 
0.67  941 


9.99  080 
9.99  078 
9.99  075 
9.99  072 
9.99  070 


0.67  878 
0.67  815 
0.67  752 
0.67  689 
0.67  627 


9.99  067 
9.99  064 
9.99  062 
9.99  059 
9.99  056 


0.67  564 
0.67  502 
0.67  439 
0.67  377 
0.67  315 


9.99  054 
9.99  051 
9.99  048 
9.99  046 
9.99  043 


0.67  253 


9.99  040 


d.  I  log  cot  I c. d 


log  tan 


log  sin 


"  eS    67    66 


6.8 
7.9 
9-1 
10.2 
"3 
22.7 
34.0 
45-3 
56.7 


6.7 
7.8 
8.9 

lO.O 
II. 2 
22.3 
33.5 

44-7 
55.8 


6.6 
7.7 
8.8 
9.9 

II.O 

22.0 
33.0 
44.0 
SS-o 


65    64    63 


6.3 

7.4 
8.4 
9.4 
10.  s 
21.0 

31. s 

42.0 
52-5 


6.5 

6.4 

7.6 

7.5 

8.7 

8.S 

Q.8 

0.6 

10.8 

10.7 

21.7 

21.3 

32.S 

32.0 

43-3 

42.7 

54.2 

53-3 

62 

61 

6.2 

6.1 

7.2 

7.1 

8.3 

8.1 

9.3 

9.2 

10.3 

10.2 

20.7 

20.3 

3I-0 

30.  s 

41.3 

40.7 

SI.7 

S0.8 

6.0 
7.0 
8.0 
9.0 

lO.O 

20.0 
30.0 
40.0 
So.o 


69 

3 

5-0 

0.3 

6.9 

0.4 

7.9 

0.4 

8.Q 

0.5 

9.8 

0.5 

19.7 

I.O 

29.5 

1.5 

39.3 

2.0 

49.2 

2.5 

Prop.  Pts. 


*i68°        258°        *348° 


78^ 
40 


Logarithms  of  the  Trigonometric  Functions 


12°                *I02^     192°     *282'     1 

/ 

log  sin 

(1. 

log  tan 

'c.d. 

log  cot 

log  COS 

Prop.  Pts. 

0 

9.31  7S8 

9.32  747 

0.67  253 

9.99  040 

60 

1 

9.31  847 

59 

9.32  810 

63 
62 

6i 
62 
63 

0.67  190 

9.99  038 

59 

2 

9.31  907 

9.32  872 

0.67  128 

9.99  035 

58 

3 

9.31  966 

59 

9.32  933 

0.67  067 

9.99  032 

57 

4 
5 

9.32  025 

59 

9.32  995 

0.67  005 

9.99  030 

56 

55 

9.32  084 

9.33  057 

0.66  943 

9.99  027 

6 

9.32  143 

59 

9.33  119 

0.66  881 

9.99  024 

54 

// 

63 

62 

61 

7 

9.32  202 

59 

9.33  ISO 

61 
62 
61 
62 

0.66  820 

9.99  022 

53 

5 

h  1 

6.1 
7.1 

8 

9.32  261 

59 
58 

9.33  242 

0.66  758 

9.99  019 

52 

7 

7.4 

7.2 

9 

9.32  319 

9.33  303 

0.66  697 

9.99  016 

51 

8 

^.4 

8.3 

8.1 

10 

9.32  378 

9.33  365 

0.66  635 

9.99  013 

50 

9 

10 

9.4 

10.  S 

9-3 

10.  s 

9.2 

10.2 

11 

9.32  437 

59 

9.33  426 

61 

0.66  574 

9.99  Oil 

49 

20 

21.0 

20.7 

20.3 

12 

9.32  495 

58 

9.33  487 

61 

0.66  513 

9.99  008 

48 

30 

31-5 

31.0 

30.5 

13 

9.32  553 

58 

9.33  548 

61 
61 
61 

0.66  452 

9.99  005 

47 

40 
50 

IZ."! 

41.3 
SI.? 

40.7 

TO.S 

14 

15 

9.32  612 

59 
S8 

9.33  609 

0.66  391 

9.99  002 

46 

9.32  670 

9.33  6/0 

0.66  330 

9.99  000 

45 

16 

9.32  728 

S8 

9.33  731 

61 

0.66  269 

9.98  997 

44 

17 

9.32  786 

58 

9.33  792 

61 

0.66  208 

9.98  994 

43 

18 

9.32  844 

58 

9.33  853 

60 
61 

0.66  147 

9.98  991 

42 

19 

9.32  902 

58 
58 

9.33  913 

0.66  087 

9-98  989 

41 

20 

9.32  960 

9.33  974 

0.66  026 

9.98  986 

40 

ou 

o» 

09 

21 

9.33  018 

58 

9.34  034 

60 

0.65  966 

9.98  983 

39 

6 

6.0 

S.Q 

5-8 

22 

9.33  075 

57 

9.34  095 

61 

0.65  905 

9.98  980 

38 

7 

7.0 

6.9 

6.8 

23 
24 

9.33  133 
9.33  190 

58 

57 
58 

9.34  155 
9.34  215 

60 
60 
61 

0.65  845 
0.65  785 

9.98  978 
9.98  975 

37 
36 

8 
9 

10 
20 
30 

40 

8.0 
9.0 

lO.O 

20.0 
30.0 
40.0 

7.9 
8.8 
9.8 
19.7 
29.S 

7.7 
8.7 
9.7 
19-3 

29.0 

38.7 

25 

9.33  248 

9.34  276 

0.65  724 

9.98  972 

35 

26 

9.33  305 

57 

9.34  336 

60 

0.65  664 

9.98  969 

34 

27 

9.33  362 

57 

9.34  396 

60 

0.65  604 

9.98  967 

33 

50 

50.0 

49.2 

48.3 

28 

9.33  420 

58 

9.34  456 

0.65  544 

9.98  964 

32 

29 
30 

9.33  477 

57 
57 

9.34  516 
9.34  576 

60 

0.65  484 

9.98  961 

31 
30 

9.33  534 

0.65  424 

9.98  958 

31 

9.33  591 

57 

9.34  635 

59 

0.65  365 

9.98  955 

29 

32 

9.33  647 

56 

9.34  695 

60 

0.65  305 

9.98  953 

28 

33 
34 
35 

9.33  704 
9.33  761 

57 
57 
57 

9.34  755 
9.34  814 
9.34  874 

60 
59 
60 

0.65  245 
0.65  186 

9.98  950 
9.98  947 

27 
26 

6 

67 

5-7 
6.6 
76 

56 

5.6 
6.5 

7  5 

56 

5-5 
6.4 
7.3 

9.33  818 

0.65  126 

9.98  944 

25 

36 

9.33  874 

56 

9.34  933 

59 

0.65  067 

9.98  941 

24 

7 
8 

37 

9.33  931 

57 

9.34  992 

59 

0.65  008 

9.98  938 

23 

9 

8.6 

8.4 

8.3 

38 

9.33  987 

56 

9.35  051 

59 

0.64  949 

9.98  936 

22 

10 

9-5 

9.3 

9.2 

39 

9.34  M3 

S6 
57 

9.35  111 

60 
59 

0.64  889 

9.98  933 

21 
20 

20 
30 
40 

ig.o 

28.  s 
38.0 

iS./ 
28.0 
37.3 

18.3 
27. 5 
36.7 

40 

9.34  100 

9.35  170 

0.64  830 

9.98  930 

41 

9.34  156 

56 

9.35  229 

59 

0.64  771 

9.98  927 

19 

SO  47.5140.7,45.0 

42 

9.34  212 

56 

9.35  288 

59 

0.64  712 

9.98  924 

18 

43 

9.34  268 

56 

9.35  347 

59 

0.64  653 

9.98  921 

17 

44 
45 

9.34  324 

56 
S6 

9.35  405 

S8 
59 

0.64  595 

9.98  919 

16 

9.34  380 

9.35  464 

0.64  536 

9.98  916 

15 

46 

9.34  436 

56 

9.35  523 

59 

0.64  477 

9.98  913 

14 

47 

9.34  491 

55 

9.35  581 

58 

0.64  419 

9.98  910 

13 

// 

S  2 

48 

9.34  547 

56 

9.35  640 

59 

0.64  360 

9.98  907 

12 

49 

9.34  602 

i3 
56 

9.35  698 

58 
59 

0.64  302 

9.98  904 

11 
10 

60 

70 
80 

.30.2 
40.2 
40.3 

50 

9.34  658 

9.35  757 

0.64  243 

9.98  901 

51 

9.34  713 

55 

9.35  815 

58 

0.64  185 

9.98  898 

9 

90 

50.3 
SO-3 

00.7 

52 

9.34  769 

56 

9.35  873 

58 

0.64  127 

9.98  896 

8 

20  I 

53 

9.34  824 

55 

9.35  931 

S8 

0.64  069 

9.98  893 

7 

30  r 

51.0 

54 

9.34  879 

55 
55 

9.35  989 

58 
58 

0.64  Oil 

9.98  890 

6 

402 
50I2 

01.3 
.5  1.7 

55 

9.34  934 

9.36  047 

0.63  953 

9.98  887 

5 

56 

9.34  989 

55 

9.36  105 

58 

0.63  895 

9.98  884 

4 

57 

9.35  044 

55 

9.36  163 

58 

0.63  837 

9.98  881 

3 

58 

9.35  099 

55 

9.36  221 

58 

0.63  779 

9.98  878 

2 

59 
60 

9.35  154 

55 
SS 

9.36  279 

S8 
57 

0.63  721 

9.98  875 

1 

9.35  209 

9.36  336 

0.63  664 

9.98  872 

0 

log  COS 

d. 

log  cot 

C.d. 

log  tan 

log  sin 

t 

Prop.  Pts. 

*i67° 

257° 

*347° 

77° 

41 


Logarithms  of  the  Trigoxometric  Functions 


13 


^103- 


193° 


^283° 


log  sin 


d. 


log  tan    c.  d. 


log  cot      log  cos 


Prop.  Pts. 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


20 

21 
22 
23 

24 


25 
26 
27 
28 
7B_ 

30 

31 
32 
33 

ii 
35 
36 
37 
38 
39 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


50 

51 

52 
53 

ii 

55 
56 
57 
58 
59 
60 


9.35 
9.35 
9.35 
9.35 
9.35 


209 
263 
318 
373 
427 


9.35 
9.35 
9.35 
9.35 
9.35 


481 
536 
590 
644 
698 


9.35 
9.35 
9.35 
9.35 
9.35 


752 
806 
860 
914 
968 


9.36 
9.36 
9.36 
9.36 
9.36 


022 
075 
129 
182 
236 


9.36 
9.36 
9.36 
9.36 
9.36 


289 
342 
395 
449 
502 


9.36 
9.36 
9.36 
9.36 
9.36 


555 
608 
660 
713 
766 


9.36 
9.36 
9.36 
9.36 
9.37 


819 
871 
924 
976 
028 


9.37 
9.37 
9.37 
9.37 
9.37 


081 
133 
185 
237 
289 


9.37 
9.37 
9.37 
9.37 
9.37 


341 
393 
445 
497 
549 


9.37 
9.37 
9.37 
9.37 
9.37 


600 
652 
703 

755 
806 


9.37 
9.37 
9.37 
9.38 
9.38 


858 
909 
960 
Oil 
062 


log  cos 

*i66° 


54 
SS 
55 
54 
54 

55 
54 
54 
54 
54 

54 
54 
54 
54 
54 

53 
54 
53 
54 
53 

53 
Si 
54 
53 
53 

53 
52 
53 
53 
53 

52 
53 
52 
52 
53 

52 
52 
52 
52 
52 

52 

52 
52 

52 

SI 

52 

51 
52 
51 

52 

51 

SI 
51 
51 
SI 

SI 
51 
51 
51 
51 

"dT 


256° 


9.36 
9.36 
9.36 
9.36 
9.36 


336 
394 
452 
509 
566 


9.36 
9.36 
9.36 
9.36 
9.36 


9.36 
9.36 
9.37 
9.37 
9.37 


624 
681 
738 
795 
852 
909 
966 
023 
080 
137 


9.37 
9.37 
9.37 
9.37 
9.37 


193 
250 
306 
363 
419 


9.37 
9.37 
9.37 
9.37 
9.37 


476 
532 
588 
644 
700 


9.37 
9.37 
9.37 
9.37 
9.37 


756 
812 
868 
924 
980 


9.38 
9.38 
9.38 
9.38 
9.38 


035 
091 
147 
202 

257 


9.38 
9.38 
9.38 
9.38 
9.38 


313 

368 
423 
479 
534 


9.38 
9.38 
9.38 
9.38 
9.38 


589 
644 
699 

754 


9.38 
9.38 
9.38 
9.39 
9.39 


863 

918. 

972 

027 

082 


9.39 
9.39 
9.39 
9.39 
9.39 


136 
190 
245 
299 
353 


9.39 
9.39 
9.39 
9.39 
9.39 
9.39 


407 
461 
515 
569 
623 
677 
log  cot 


•^346° 


58 

S8 
57 
57 
58 

57 
57 
57 
57 
57 

57 
57 
57 
57 
56 

57 
56 
57 
56 
57 

S6 
56 
S6 
S6 
56 

S6 
56 
56 
56 
55 

56 
S6 
55 
55 
56 

55 
55 
56 
55 
$S 

55 
55 
55 
54 
55 

55 
>54 
55 
55 
54 

54 
55 
54 
54 
54 

54 
54 
54 
54 
54 

cTd! 


0.63  664 
0.63  606 
0.63  548 
0.63  491 
0.63  434 
0.63  376 
0.63  319 
0.63  262 
0.63  20i 
0.63  148 


9.98  872 
9.98  869 
9.98  867 
9.98  864 
9.98  861 
9.98  858 
9.98  855 
9.98  852 
9.98  849 
9.98  846 


0.63  091 
0.63  034 
0.62  977 
0.62  920 
0.62  863 


9.98  843 
9.98  840 
9.98  837 
9.98  834 
9.98  831 


0.62  807 
0.62  750 
0.62  694 
0.62  637 
0.62  581 


9.98  828 
9.98  825 
9.98  822 
9.98  819 
9.98  816 


0.62  524 
0.62  468 
0.62  412 
0.62  356 
0.62  300 


9.98  813 
9.98  810 
9.98  807 
9.98  804 
9.98  801 


0.62  244 
0.62  188 
0.62  132 
0.62  076 
0.62  020 


9.98  798 
9.98  795 
9.98  792 
9.98  789 
9.98  786 


0.61 
0.61 
0.61 
0.61 
0.61 


965 
909 
853 
798 

743 


9.98  783 
9.98  780 
9.98  777 
9.98  774 
9.98  771 


0.61  687 

0.61  632 

0.61  577 

0.61  521 

0.61  466 


9.98  768 
9.98  765 
9.98  762 
9.98  759 
9.98  756 


0.61  411 

0.61  356 

0.61  301 

0.61  246 

0.61  192 


9.98  753 
9.98  750 
9.98  746 
9.98  743 
9.98  740 


0.61  137 
0.61  082 
0.61  028 
0.60  973 
0.60  918 


9.98  737 

9.98  734 

9.98  731 

9.98  728 

9.98  725 


0.60  864 
0.60  810 
0.60  755 
0.60  701 
0.60  647 


9.98  722 
9.98  719 
9.98  715 
9.98  712 
9.98  709 


0.60  593 
0.60  539 
0.60  485 
0.60  431 
0.60  377 
0.60  323 


9.98  706 
9.98  703 
9.98  700 
9.98  697 
9.98  694 
9.98  690 


log  taiil  log  sin 

76° 

42 


58 

67 

5.8 
6.8 

5.7 
6.5 

7.7 
8.7 

7.6 
8.6 

Q-7 
19.3 
29.0 
38.7 
48.3 

9.5 
19.0 
28.5 
38.0 
47.5 

// 

55 

64 

6 

.5.5 

5.4 

7 

6.4 

6.3 

8 

7.,S 

7.2 

9 

8.3 

8.1 

10 

9.2 

Q.O 

20 

18.3 

18.0 

30 

27.5 

27.0 

40 

36.7 

36.0 

SO 

45.8 

45.0 

66 

5.6 
6.5 
7.5 
8.4 
9.3 
[8.7 
28.0 
37.3 
46.7 


63 

5-3 
6.2 

7.1 
8.0 
8.8 
17.7 
26.S 
35.3 
44.2 


62    51     4 


5-2 

6.1 

6.9 

7.8 

8.7 

17.3 

26.0 

34.7 

43-3 


5.1 

6.0 

6.8 

7-7 

8.5 

17.0 

25-5 

34.0 

42.5 


0.4 
0-5 
0.5 
0.6 
0.7 
1-3 
2.0 
2.7 
3.3 


" 

3 

6 

0.3 

7 

0.4 

8 

0.4 

9 

o-S 

10 

o.S 

20 

I.O 

30 

1. 5 

4& 

2.0 

50 

2.5 

0.2 
0.2 

0.3 
0.3 
0.3 
0.7 

1.0 
1-3 
1.7 


Prop.  Pts. 


Logarithms  of  the  TitiGoxoMEXKic  Fuxotions 


14° 

*i 

04"    194^    *284^    1 

0 

log  sin 

d. 

log  tau 

c.  d. 

log  cot 

log  cos 

d. 

Prop.  Pts. 

9. 38  368 

9.39  677 

0.60  323 

9.98  690 

60 

1 

9.38  418 

so 

9.39  731 

54 

0.60  269 

9.98  687 

3 

59 

2 

9.38  469 

51 

9.39  785 

54 

0.60  215 

9.98  684 

3 

58 

3 

9.38  519 

SO 

9.39  838 

S3 

0.60  162 

9.98  681 

3 

57 

4 

5 

9.38  570 

SI 
50 

9.39  892 

S4 
S3 

0.60  108 

9.98  678 

3 
3 

56 

55 

9.38  620 

9.39  945 

0.60  055 

9.98  675 

6 

9.38  670 

SO 

9.39  999 

54 

0.60  001 

9.98  671 

4 

54 

"  54 

63  !  52     1 

7 

9.38  721 

SI 

9.40  052 

S3 

0.59  948 

9.98  668 

3 

53 

6  <;  .^ 

.J., 

8 

9.38  771 

50 

9.40  106 

54 

0.59  894 

9.98  665 

3 

52 

7  6.3I  6.2I  6.1     1 

9 

9.38  821 

so 
50 

9.40  159 

S3 
53 

0.59  841 

9.98  662 

3 
3 

51 

8  7. 

!   7.1  6.9       1 
8.0I  7.8       1 

3  8.8i  8.7     1 

10 

9.38  871 

9.40  212 

0.59  788 

9.98  659 

50 

10  g.c 

11 

9.38  921 

50 

9.40  266 

54 

0.59  734 

9.98  656 

3 

49 

2o!i8.o!i7.7'i7.3 

12 

9.38  971 

50 

9.40  319 

5i 

0.59  681 

9.98  652 

4 

48 

30  27.0  26.5  26.0 
4036.0  3S.3'34.7 
50  45.044. 2  43.3 

13 

9.39  021 

so 

9.40  372 

S3 

0.59  628 

9.98  649 

3 

47 

14 

9.39  071 

SO 
SO 

9.40  425 

53 
S3 

0.59  575 

9.98  646 

3 
3 

46 
45 

IS 

9.39  121 

9.40  478 

0.59  522 

9.98  643 

16 

9.39  170 

49 

9.40  531 

53 

0.59  469 

9.98  640 

3 

44 

17 

9.39  220 

50 

9.40  584 

53 

0.59  416 

9.98  636 

4 

43 

18 

9.39  270 

SO 

9.40  636 

52 

0.59  364 

9.98  633 

3 

42 

19 

9.39  319 

49 

9.40  689 

53 
53 

0.59  311 

9.98  630 

3 

41 

"  51 

|50  49 

20 

9.39  369 

9.40  742 

0.59  258 

9.98  627 

40 

21 

9.39  418 

49 

9.40  79i 

S3 

0.59  205 

9.98  623 

4 

39 

6  s. 

i  S.o  4.9 

22 

9.39  467 

49 

9.40  847 

52 

0.59  153 

9.98  620 

3 

38 

7  6.0  5.81  5.7     1 

8  6.8  6.7j  6.5     1 
g  7.7  7.1;!  7./1     1 

23 

9.39  517 

SO 

9.40  900 

53 

0.59  100 

9.98  617 

3 

37 

24 

9.39  566 

49 

9.40  952 

52 
53 

0.59  048 

9.98  614 

3 

36 

10  8. 

5  8.3J  8.2      1 

25 

9.39  615 

9.41  005 

0.58  995 

9.98  610 

35 

30  25.5  25.0  24.5 

26 

9.39  664 

49 

9.41  057 

52 

0.58  943 

9.98  607 

3 

34 

40  34.0  33-3  32-7 

27 

9.39  713 

49 

9.41  109 

52 

0.58  891 

9.98  604 

3 

33 

50  42.5  41-7  40.8 

28 

9.39  762 

49 

9.41  161 

52 

0.58  839 

9.98  601 

3 

32 

29 

9.39  811 

49 
49 

9.41  214 

53 
52 

0.58  786 

9.98  597 

4 

31 

30 

9.39  860, 

9.41  266 

0.58  734 

9.98  594 

30 

31 

9.39  909" 

49 

9.41  318 

52 

0.58  682 

9.98  591 

3 

29 

32 

9.39  958 

49 

9.41  370 

52 

0.58  630 

9.98  588 

3 

28 

33 

9.40  006 

48 

9.41  422 

52 

0.58  578 

9.98  584 

4 

27 

"  i 

18  47 

34 

9.40  055 

49 
48 

9.41  474 

52 
52 

0.58  526 

9.98  581 

3 

26 

35 

9.40  103 

9.41  526 

0.58  474 

9.98  578 

25 

7 
8 

4.8  4.7 
5.4  0.3 

36 

9.40  152 

49 

9.41  578 

52 

0.58  422 

9.98  574 

4 

24 

37 

9.40  200 

48 

9.41  629 

51 

0  58  371 

9.98  571 

3 

23 

9 

7.2  7.0 

38 

9.40  249 

49 

9.41  681 

52 

0.58  319 

9.98  568 

3 

22 

10 

8.0  7.8 
6.0  15.7 
40  23.S 

2.or?i.3 

39 
40 

9.40  297 

48 
49 

9.41  733 

52 

51 

0.58  267 

9.98  565 

3 
4 

21 
20 

30,2 
40|3 

9.40  346 

9.41  784 

0.58  216 

9.98  561 

41 

9.40  394 

48 

9.41  836 

52 

0.58  164 

9.98  558 

3 

19 

SO;4i 

42 

9.40  442 

48 

9.41  887 

51 

0.58  113 

9.98  555 

3 

18 

43 

9.40  490 

48 

9.41  939 

52 

0.58  061 

9.98  551 

4 

17 

44 

45 

9.40  538 

48 

48 

9.41  990 

51 
51 

0.58  010 

9.98  548 

3 

3 

16 
15 

9.40  586 

9.42  041 

0.57  959 

9.98  545 

46 

9.40  634 

48 

9.42  093 

52 

0.57  907 

9.98  541 

4 

14 

47 

9.40  682 

48 

9.42  144 

51 

0.57  856 

9-98  538 

3 

13 

// 

4   S 

48 

9.40  730 

48 

9.42  195 

51 

0.57  805 

9.98  535 

3 

12 

6 

7 
8 

49 
50 

9.40  778 

48 

47 

9.42  246 

51 
51 

0.57  754 

9.98  531 

4 
3 

11 
10 

o.s  0.4 
0.5  0.4 

9.40  825 

9.42  297 

0.57  703 

9.98  528 

51 

9.40  873 

48 

9.42  348 

51 

0.57  652 

9.98  525 

3 

9 

9 

0.7  o.s 
1.3  i.o 

52 

9.40  921 

48 

9.42  399 

SI 

0.57  601 

9.98  521 

4 

8 

20 

53 

9.40  968 

47 

9.42  450 

SI 

0.57  550 

9.98  518 

3 

7 

30 

2.0  1.5 

54 

55 
56 

9.41  016 

48 
47 

48 

9.42  501 

SI 
SI 

0.57  499 

9.98  515 

3 

4 

3 

6 

5 
4 

40 
50 

3-3  2.S 

9.41  063 
9.41  111 

9.42  552 
9.42  603 

0.57  448 

9.98  511 
9.98  508 

SI 

0.57  397 

57 

9.41  158 

47 

9.42  653 

SO 

0.57  347 

9.98  505 

3 

3 

58 

9.41  205 

47 

9.42  704 

51 

0.57  296 

9.98  501 

4 

2 

59 

9.41  252 

47 

48 

9.42  755 

51 

0.57  245 

9.98  498 

3 

1 

60 

9.41  300 

9.42  805 

0.57  195 

9.98  494 

0 

log  COS 

"m 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*i65° 

255^ 

*345" 

75° 

1 

43 


Logarithms  of  the  Trigonometric  Functions 


15 


405^ 


195° 


*285° 


log  sin 


10 

11 

12 
13 

it 
15 
16 
17 
18 
29 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 
32 
33 

it 
35 
36 
37 
38 
39^ 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49^ 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.41 
9.41 
9.41 
9.41 
9.41 
9.41 
9.41 
9.41 
9.41 
9.41 
9.41 
9.41 
9.41 
9.41 
9.41 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.42 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.43 
9.44  034 

log  COS 

*i64° 


300 
347 
394 
441 

488 

535 
582 
628 
675 
722 
768 
81i 
861 
908 
954 
001 
047 
093 
140 
186 
232 
278 
324 
370 
416 
461 
507 
553 
599 
644 
690 
735 
781 
826 
872 
917 
962 
008 
053 
098 
143 
188 
233 
278 
323 

367 
412 
457 
502 
546 
591 
635 
680 
724 
769 
813 
857 
901 
946 
990 


d. 


47 
47 
47 
47 
47 

47 
46 
47 
47 
46 

47 
46 
47 
46 
47 

46 
46 
47 
46 
46 

46 
46 
46 
46 

45 

46 
46 
46 

45 
46 

45 
46 
45 
46 
45 

45 
46 
45 
45 
45 

45 
45 
45 
45 
44 

45 
45 
45 
44 
45 

44 
45 
44 
45 
44 

44 
44 
45 
44 
44 

"dT 


tan  c.  d.    log  cot      log  cos 


9.42  805 
9.42  856 
9.42  906 

9.42  957 

9.43  007 


9.43  057 
9.43  108 
9.43  158 
9.43  208 
9.43  258 


9.43 
9.43 


308 
358 


9.43  408 
9.43  458 
9.43  508 


9  43  558 
9.43  607 
9.43  657 
9.43  707 
9.43  756 


9.43  806 
9.43  855 
9.43  905 

9.43  954 

9.44  004 


9.44  053 
9.44  102 
9.44  151 
9.44  201 
9.44  250 


9.44  299 
9.44  348 
9.44  397 
9.44  446 
9.44  495 


9.44  544 
9.44  592 
9.44  641 
9.44  690 
9.44  738 


9.44  787 
9.44  836 
9.44  884 
9.44  933 
9.44  981 


9.45  029 
9.45  078 
9.45  126 
9.45  174 
9.45  222 


9.45  271 
9.45  319 
9.45  367 
9.45  415 
9.45  463 


254° 


9.45  511 

9.45  559 

9.45  606 

9.45  654 

9.45  702 

9.45  750 

log  cot 


^344° 


51 
so 
51 
SO 
SO 

51 
50 
50 
SO 
SO 

SO 
50 
50 
50 
SO 

49 

SO 
SO 
49 
SO 

49 
50 
49 
SO 
49 

49 
49 
SO 
49 
49 

49 
49 
49 
49 
49 

48 
49 
49 
48 
49 

49 


48 


48 


c.  d. 


57  195 
57  144 
57  094 
57  043 
56  993 


56  943 
56  892 
56  842 
56  792 
56  742 


56  692 
56  642 
56  592 
56  542 
56  492 


56  442 
56  393 
56  343 
56  293 
56  244 


56  194 
56  145 
56  095 
56  046 
55  996 


55  947 

55  898 
55  849 
55  799 
55  750 


55  701 
55  652 
55  603 
55  554 
55  505 


55  456 
55  408 
55  359 
55  310 
55  262 


55  213 
55  164 
55  116 
55  067 
55  019 


0.54  971 
0.54  922 
0.54  874 
0.54  826 
0.54  778 


0.54  729 
0.54  681 
0.54  633 
0.54  585 
0.54  537 


0.54  489 
0.54  441 
0.54  394 
0.54  346 
0.54  298 
0.54  250 

log  tan 

74° 


9.98  494 
9.98  491 
9.98  488 
9.98  484 
9.98  481 


9.98  477 
9.98  474 
9.98  471 
9.98  467 
9.98  464 


9.98  460 
9.98  457 
9.98  453 
9.98  450 
9.98  447 


9.98  443 
9.98  440 
9.98  436 
9.98  433 
9.98  429 


9.98  426 
9.98  422 
9.98  419 
9.98  415 
9.98  412 


9.98  409 
9.98  405 
9.98  402 
9.98  398 
9.98  395 


9.98  391 
9.98  388 
9.98  384 
9.98  381 
9.98  377 


9.98  373 
9.98  370 
9.98  366 
9.98  363 
9.98  359 


9.98  356 
9.98  352 
9.98  349 
9.98  345 
9.98  342 


9.98  338 
9.98  334 
9.98  331 
9.98  327 
9.98  324 


9.98  320 
9.98  317 
9.98  313 
9.98  309 
9.98  306 


9.98  302 
9.98  299 
9.98  295 
9.98  291 
9.98  288 
9.98  284 

log  sin 


d. 


Prop.  Pts. 


It 

51 

60 

6 

7 
8 

S-i 

6.0 

6.8 

S-o 
5.8 
6.7 

9 
10 
20 

7.7 
8.S 
17.0 

7.5 
8.3 
16.7 

30 
40 
SO 

25.5 
34.0 
42-5 

25.0 
33-3 
41.7 

48 

47 

4.8 

4-7 

5.6 

5.5 

6.4 

6.3 

7.2 

7.0 

8.0 

7.8 

16.0 

IS- 7 

24.0 

23-5 

32.0 

31-3 

40.0 

39-2 

49 

4.9 

5.7 
6.S 
7.4 
8.2 
16.3 

24-5 

32.7 
40.8 


46 

4.6 

5-4 

6.1 

6.9 

7-7 

15.3 

23.0 

30.7 

38.3 


// 

45 

6 

4-5 

7 

5-3 

« 

6.0 

0 

6.8 

10 

7-5 

20 

iS.o 

30 

22.5 

40 

30.0 

SO 

37-S 

44 

4.4 
S-i 
5-9 
6.6 
7.3 
14-7 
22.0 
29.3 
36.7 


» 

4 

6 

0.4 

7 

0.5 

8 

o.S 

9 

0.6 

10 

0.7 

20 

1.3 

30 

2.0 

40 

2.7 

50 

3-3 

Prop.  Pts. 


4.4. 


LOQAKITHMS    OF    THE    TrIGOXOMETRIC    FUNCTIONS 


16 


*io6= 


196^ 


*2i 


log  slu        (1. 


log  tan    c.  (1.    log  cot 


log  cos       (1. 


Prop.  P(s. 


0 

1 
2 
3 

5 

6 

7 

8 

9^ 

10 

11 

12 

13 

14 

15 

16 

17 

18 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 

15" 
36 
37 
38 
39 

40 

41 
42 
43 
44 
45 
46 
47 
48 
jt9 

50 

51 
52 
53 

ii 

55 
56 
57 

58 

60 


9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.44 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.45 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46  .594 


034 
078 
122 
166 
210 
253 
297 
341 
385 
428 
"472" 
516 
559 
602 
646 
689 
733 
776 
819 
862 
905 
948 
992 
035 
077 
120 
163 
206 
249 
292 

334^ 

377 

419 

462 
504 

547 
589 
632 
674 
716 
758 
801 
843 
885 
927 
969 
Oil 
053 
095 
136 
178 
220 
262 
303 
345 
38b 
428 
469 
511 
552 


44 
44 
44 
43 
44 

44 
43 
43 
44 
43 

44 
43 
43 
43 
43 

43 

44 
43 
42 
43 

43 
43 
43 
43 


9.45 
9.45 


750 
797 


9.45   84i 


9.45 
9.45 


892 
940 


9.45  987 

9.46  035 
9.46  082 
9.46  130 
9.46  177 


9.46  224 
9.46  271 
9.46  319 
9.46  366 
9.46  413 


9.46  460 
9.46  507 
9.46  554 
9.46  601 
9.46  648 


9.46  694 
9.46  741 
9.46  788 
9.46  835 
9.46  881 


9.46  928 

9.46  975 

9.47  021 
9.47  068 
9.47  114 


9.47  160 
9.47  207 
9.47  253 
9.47  299 
9.47  346 


9.47  392 
9.47  438 
9.47  484 
9.47  530 
9.47  576 


9.47 
9.47 
9.47 
9.47 


622 
668 
714 
760 


9.47  806 


9.47  852 
9.47  897 
9.47  943 

9.47  989 

9.48  035 


9.48 
9.48 
9.48  262 


9.48  307 
9.48  353 
9.48  398 
9.48  443 
9.48  489 


9.48  534 


47 
47 
47 

47 
47 
47 
47 
46 

47 
47 
47 
46 
47 

47 
46 
47 
46 
46 

47 
46 
46 
47 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

45 
46 
46 
46 

45 

46 
45 
46 
45 
45 

46 
45 
45 
46 
45 


.54  250 

.54  203 

.54  155 

.54  108 

54  060 


9.98  284 
9.98  281 
9.98  277 
9.98  273 
9.98  270 


54  013 
53  965 
53  918 
53  870 
53  823 


9.98  266 

9.98  262 

9.98  259 

9.98  255 

9.98  25] 


53  776 

53  729 

53  681 

53  634 

.53  587 


9.98  248 
9.98  244 
9.98  240 
9.98  237 
9.98  233 


53  540 
53  493 
53  446 
53  399 
53  352 


9.98  229 

9.98  226 

9.98  222 

9.98  218 

9.98  215 


53  306 
53  259 
53  212 
53  165 
53   119 


9.98  211 
9.98  207 
9.98  204 
9.98  200 
9.98  196 


53  072 
53  025 
52  979 
52  932 

52  886 


9.98  192 
9.98  189 
9.98  185 
9.98  181 
9.98  177 


52  840 
52  793 
52  747 
52  701 
52  654 


9.98 
9.98 
9.98 
9.98 
9.98 


174 
170 
166 
162 
159 


52  608 
52  562 
52  516 
52  470 
52  424 


9.98  155 

9.98  151 

9.98  147 

9.98  144 

9.98  140 


52  378 
52  332 
52  286 
52  240 
52  194 


9.98  136 
9.98  132 
9.98  129 
9.98  125 
9.98  121 


52  148 
52  103 
52  057 
52-011 
51  965 


9.98  117 
9.98  113 
9.98  110 
9.98  106 
9.98  102 


51  920 
51  874 
51  829 
51  783 
51  738 
51  693 
51  647 
51  602 
51  557 
51  511 


9.98  098 
9.98  094 
9.98  090 
9.98  087 
9.98  083 


9.98  079 
9.98  075 
9.98  071 
9.98  067 
9.98  063 


51  466 


9.98  060 


log  COS      d. 


log  cot    c.  d.    log  tan 


log  sin      d. 


"  48    47    46 


16.0  15.7 
24.0  23.5 
32.031.3 
40.0 1 39. 2 


45 


o    4-5 

7  5.3 

8  6.0 

9  6.8 
10  7-5 
20  15.0 

30  22.5 
40130.0 

50;37.5 


44 

4.4 

5-1 

5.9 

6.6 

7.3 

14.7 

22.0 

29.3 

36.7 


4.6 
5-4 
6.1 
6.9 

7-7 
15-3 
23.0 
30.7 
38.3 


43 

4.3 

5.0 

5-7 

6.4 

7.2 

14.3 

21.S 

28.7 

35.8 


42    41 


61  4.2 

7  4-9 

8  5.6 
9!  6.3 

10,  7.0 
20  14.0 
30' 21.0 
40  28.0 
50;3S-o 


4.1 
4.8 

6.2 

6.8 
13.7 
20.  s 
27.3 
34-2 


// 

4 

6 

0.4 

7 

o.S 

8 

o.s 

9 

0.6 

10 

0.7 

20 

1-3 

.SO 

2.0 

40 

2.7 

SO 

3.3 

0.3 

0.4 

0.4 

O.S 
o.S 
i.o 
i-S 
2.0 

2-S 


Prop.  Pts. 


^163^ 


253 


^343' 


73° 


45 


Logarithms  of  the  Trigoxometric  Functions 


w 


Hof 


197- 


1^287° 


loff  sin 


log  tan 


c.  (1. 


log  cot 


log  COS 


d. 


Prop.  Pts. 


9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 
9.46 


9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.47 
9.48 
9.48 
9.48 
9.48 
9.48 


800 
841 
882 
923 
964 
005 
045 
086 
127 
168 
209 
249 
290 
330 
371 


9.48 
9-48 
9.48 
9.48 
9.48 
9.48 
9.48 
9.48 
9.48 
9.48 


411 

450 
490 
529 
568 
607 
647 
686 
725 
764 
803 
842 
881 
920 
959 


9.48  998 
log  cos   d. 


9.48  534 
9.48  579 
9.48  624 
9.48  669 
9.48  714 
9.48  759 
9.48  804 
9.48  849 
9.48  894 
9.48  939 


9.48  984 

9.49  029 
9.49  073 
9.49  118 
9.49  163 


9.49  207 
9.49  252 
9.49  296 
9.49  341 
9.49  385 


9.49  430 
9.49  474 
9.49  519 
9.49  563 
9.49  607 


9.49  652 
9.49  696 
9.49  740 
9.49  784 
9.49  828 


9.49  872 
9.49  916 

9.49  960 

9.50  004 
9.50  048 


9.50  092 
9.50  136 
9.50  180 
9.50  223 
9.50  267 


9.50  311 
9.50  355 
9.50  398 
9.50  442 
9.50  485 


9.50  529 
9.50  572 
9.50  616 
9.50  659 
9.50  703 


9.50  746 
9.50  789 
9.50  833 
9.50  876 
9.50  919 


*l62° 


252" 


9.50  962 

9.51  005 
9.51  048 
9.51  092 
9.51  135 
9.51  178 

log  cot 

*342° 


45 
45 

4S 
45 
45 

45 

45 
45 
45 
45 

45 
44 
45 
45 
44 

45 
44 
45 
44 
4J 

44 
45 
44 
44 
45 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
43 
44 
44 

44 
43 
44 
43 
44 

43 
44 
43 
44 
43 

43 

44 
43 
43 
43 

43 
43 
44 
43 
43 

c~ 


51  466 
51  421 
51  376 
51  331 
.51  286 


9.98  060 
9.98  056 
9.98  052 
9.98  048 
9.98  044 


51  241 
51  196 
51  151 
51  106 
51  061 


9.98  040 
9.98  036 
9.98  032 
9.98  029 
9.98  025 


51  016 
50  971 
50  927 
50  882 
50  837 


9.98  021 
9.98  017 
9.98  013 
9.98  009 
9.98  005 


50  793 
50  748 
50  704 
50  659 
50  615 


9.98  001 
9.97  997 
9.97  993 
9.97  989 
9.97  986 


50  570 
50  526 
50  481 
50  437 
50  393 


9.97  982 
9.97  978 
9.97  974 
9.97  970 
9.97  966 


50  348 
50  304 
50  260 
50  216 
50  172 


9.97  962 
9.97  958 
9.97  954 
9.97  950 
9.97  946 


50  128 
50  084 
50  040 
49  996 
49  952 


9.97  942 
9.97  938 
9.97  934 
9.97  930 
9.97  926 


49  908 
49  864 
49  820 
49  777 
49  733 


9.97  922 
9.97  918 
9.97  914 
9.97  910 
9.97  906 


49  689 
49  645 
49  602 
49  558 
49  515 


9.97  902 
9.97  898 
9.97  894 
9.97  890 
9.97  886 


49  471 
49  428 
49  384 
49  341 
49  297 


9.97  882 
9.97  878 
9.97  874 
9.97  870 
9.97  866 


49  254 
49  211 
49  167 
49  124 
49  081 


9.97  861 
9.97  857 
9.97  853 
9.97  849 
9.97  845 


49  038 
48  995 
48  952 
48  908 
48  865 
48  822 

log  t<an 

72° 


9.97  841 
9.97  837 
9.97  833 
9.97  829 
9.97  825 
9.97  821 


log  sin      d. 


" 

46 

44 

6 

4-5 

4.4 

7 

5.3 

5-1 

8 

6.0 

5.Q 

0 

6.8 

6.6 

10 

7.5 

7.3 

20 

15.0 

14.7 

30 

22.5 

22.0 

40 

30.0 

20.3 

50 

37.S 

36.7 

43 

4.3 

5.0 

5.7 

6.4 

7.2 

14.3 

21. S 

28.7 

35.8 


42 

4.2 

4.9 

5.6 

6.3 

7.0 

14.0 

21.0 

28.0 

35-0 


41 

4.1 

4.8 

5-1 

6.2 

6.8 

13.7 

20.5 

27.3 

34.2 


"  40    39 


4.0 

4-7 

5.3 

6.0 

6.7 

13-3 

20.0 

26.7 

33-3 


3-9 

4.6 

5-2 

5-9 

6.5 

13.0 

19.5 

26.0 

32.S 


5  I  4 


o.6jo.5'o.4 

0.7 

0.8 

0.8 

1.7 

2.5 

3.3 

4.2 


o.S|o.4 
0.6  0.5 
0.7  o.s 


Prop.  Pts. 


46 


Logarithms  of  the  Trigonometric  Functions 


18° 


*io8^ 


*288° 


log  sin      d. 


log  tan    c.  d.    log  cot 


log  cos 


d. 


Prop.  Pts. 


10 

11 
12 
13 

ii 

15 
16 

17 
18 

}1 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29_ 
30 
31 
32 
33 

IL 
35 
36 
37 
38 
39_ 
40 
41 
42 
43 
44 
45 
46 
47 
48 
j9_ 
50 
51 
52 
53 

ii 

55 
56 
57 
58 

ii 
60 


9.48 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.49 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9.50 

9. SO 

9.50 

9.50 

9.50 

9.50 

9:50" 

9.50 

9.  ,50 

9.51 

9.51 

9.51 

9.51 

9.51 

9.51 

9.51 


998  I 

037 

076 

115  ! 
153  I 
192  '' 
231  ! 
269  ' 
308  I 
347  j 
385  I 
424 
462 
500  ' 

539  ; 

577 
615  ' 
654  : 
692 
730 

76S~: 

806  ■ 

844 

882 

920 

958 

996  i 

034  < 

072  i 

110  i 

148  I 

185 

223 

261 

298 

33o 

374 

411, 

449 

486  ' 

523 

561  i 

598  ! 

63S 

673  ' 

710 

747  ! 

784  1 

821 

858  j 

896  ' 

933 

970 

007 

043 

080 

117 

154 

191 

227 


9.51   264 


39 
39 
39 
38 
39 

39 
38 
39 
39 
38 

39 
38 
38 
39 
38 

38 
39 
38 
38 
38 

38 
28 
38 
38 
38 

38 
38 
38 
38 
38 

37 
38 
38 
37 
38 

38 
37 
38 
37 
37 

38 
37 
37 
38 
37 

37 
37 
37 
37 
38 

37 
37 
37 
36 
37 

37 
37 
37 
36 
37 


log  COS 


*i6i'^ 


9.51 
9.51 
9.51 
9.51 
9.51 


178 
221 
264 
306 
349 


9.51 
9.51 
9.51 
9.51 
9.51 


392 
435 
478 
520 
563 


9.51 
9.51 
9.51 
9.51 
9.51 


606 
648 
691 
734 
776 


9.51 
9.51 
9.51 
9.51 
9.51 


819 
861 
903 
946 

988 


9.52 
9.52 
9.52 
9.52 
9.52 


9.52 
9.52 
9.52 
9.52 
9.52 


9.52 
9.52 
9.52 
952 
9.52 


452 
494 
536 
578 
620 


9.52 
9.52 
9.52 
9.52 
9.52 


661 
703 

745 
787 
829 


9.52 
9.52 
9.52 
9.52 
9.53 


870 
912 
953 
995 
037 


9.53 
9.53 
9  53 
9.53 
9.53 


078 
120 
161 
202 
244 


9.53 
9.53 
9.53 
9.53 
9.53 


285 
327 
368 
409 
450 


9.53 
9.53 
9.53 
9.53 
9.53 


492 
533 
574 
615 
656 


031 
073 
115 
157 
200 


242 
284 
326 
368 
410 


9.53  697 


43 
43 
42 
43 
43 

43 

43 
42 
43 
43 

42 
43 
43 
42 
43 

42 
42 
43 

42 
43 


42 
42 
42 
42 
41 

42 
41 
42 
42 
41 

42 
41 
41 
42 
4X 

42 
41 
41 
41 
42 

41 
41 
41 
41 
41 


0.48  822 
0.48  779 
0.48  736 
0.48  694 
0.48  651 


9.97  821 

9.97  817 

9.97  812 

9.97  808 

9.97  804 


0.48  608 

0.48  565 

0.48  522 

0.48  480 

0.48  437 


9.97 
9.97 
9.97 
9.97 
9.97 


0.48  394 

0.48  352 

0.48  309 

0.48  266 

0.48  224 


9.97 
9.97 
9.97 
9.97 
9.97 


0.48 
0.48 
0.48 
0.48 


181 
139 
097 
054 


0.48  012 


0.47 
0.47 
0.47 
0.47 
0.47 


969 
927 
885 
843 
800 


0.47 
0.47 
0.47 
0.47 
0.47 
0.47 
0.47 
0.47 
0.47 
0.47 


758 
716 
674 
632 
590 
548 
506 
464 
422 
380 


0.47 
0.47 
0  47 
0.47 
0.47 


339 
297 

255 
213 
171 


0.47  130 

0.47  088 

0.47  047 

0.47  005 

0.46  963 


9.97 
9.97 
9.97 
9.97 
9.97 


0.46 
0.46 
0.46 
0.46 
0.46 


922 
880 
839 
798 
756 


9.97 
9.97 
9.97 
9.97 
9.97 


0.46 
0.46 
0.46 
0.46 
0.46 


715 
673 
632 
591 

550 


9.97 
9.97 
9.97 
9.97 
9.97 


0.46  .508 

0.46  467 

0.46  426 

0.46  385 

0.46  344 


9.97 
9.97 
9.97 
9.97 
9.97 


800 
796 
792 
788 
784 


779 
775 
771 
767 
763 


9.97 
9.97 
9.97 
9.97 
9.97 


759 
754 
750 
746 
742 


9.97  738 

9.97  734 

9.97  729 

9.97  725 

9.97  721 


9.97  717 

9.97  713 

9.97  708 

9.97  704 

9.97  700 

9.97  696 

9.97  691 

9.97  687 

9.97  683 

9.97  679 


9.97  674 
9.97  670 
9.97  666 
9.97  662 
9.97  657 


653 
649 
645 
640 
636 


632 
628 
623 
619 
615 


610 
606 
602 
597 
593 


589 
584 
580 
576 
571 


0.46  303 


9.97  567 


log  cot    c.  d.    log  tan      log  sin 


d. 


43 

4.3 

42 

4.2 

S-o 

4-9 

H-1 

^.b 

6.4 

6.3 

7.2 

7.0 

14-3 

14.0 

21. S 

21.0 

28.7 

28.0 

35.8 

35.0 

4.1 
4.8 

S-5 
6.2 
6.8 
13.7 
20.  s 
27-3 
34-2 


" 

39 

38 

6 

3-9 

3.8 

7 

4.0 

4-4 

8 

5-2 

.S.! 

9 

,S.9 

.■;.7 

10 

<>..■; 

6.3 

20 

13.0 

12.7 

30 

19.S 

19.0 

40 

26.0 

25.3 

SO 

32.S 

31.7 

37 

3-7 
4.3 
4.9 
S.6 
6.2 
12.3 
18.S 
24.7 


// 

36 

5 

6 

3.6 

0.5 

7 

4.2 

0.6 

8 

4.8 

0.7 

9 

.S.4 

0.8 

10 

6.0 

0.8 

20 

12.0 

1.7 

30 

18.0 

2.S 

40 

24.0 

3.3 

SO 

30.0 

4.2 

0.4 

o.S 
o.S 
0.6 
0.7 
1-3 
2.0 
2.7 
3-3 


Proi».  Pts. 


251^ 


^341" 


71 


47 


Logarithms  of  the  Trigonometric  Functions 


19° 


*io9'' 


199- 


*289" 


loj?  sill 


(1. 


log  tan   c.  <1.    log  cot 


log  cos 


d. 


Prop.  Pts. 


0 

1 
2 
3 

5 

6 

7 

8 

9_ 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
"55" 
56 
57 
58 
59 
60 


9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.51 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.52 
9.53 
9.53 
9.53 
9.53 
9.53 
9.  .53 
9.53 
9.53 
9.53 
9.53 
9.53 
9.53   405 


264 
301 
338 
374 
411 

447 
484 
520 
557 
593 
629 
666 
702 
738 
774 
811 
847 
883 
919 
955 
991 
027 
063 
099 
135 
171 
207 
242 
278 
314 
350 
385 
421 
456 
492 

527 
563 
598 
634 
669 
705 
740 
775 
811 
846 
881 
916 
951 
986 
021 
056 
092 
126 
161 
196 
2ii 
266 
301 
336 
370 


37 
37 
36 
37 
36 

37 
36 
37 
36 
36 

37 
36 
36 
36 
37 

36 
36 
36 
36 
36 

36 
36 
36 
36 
36 

36 
35 
36 
36 
36 

35 
36 
35 
36 
35 

36 
35 
36 
35 
36 

35 
35 
36 
35 
35 

35 
35 
35 
35 
35 

36 
34 
35 
35 
35 

35 
35 
35 
34 
35 


9.53 
9.53 
9.53 
9.53 
9.53 


697 

738 
779 
820 
861 


9.53 
9.53 
9.53 
9.54 
9.54 


902 
943 
984 
025 
065 


9.54 
9.54 
9.54 
9.54 
9.54 


106 
147 
187 
228 
269 


9.54 
9.54 
9.54 
9.54 
9.54 


309 
350 
390 
431 
471 


9.54 
9.54 
9.54 
9.54 
9.54 


512 
552 
593 
633 
673 


9.54 
9.54 
9.54 
9.-54 
9.54 


714 

754 
794 
835 
875 


9.54 
9.54 
9.54 
9.55 
9.55 


915 
955 
995 
035 
075 


9.55 
9.55 
9.55 
9.55 
9.55 


115 

155 
195 
235 

275 


9.55 
9.55 
9.55 
9.55 
9.55 


315 
355 
395 
434 
474 


9.55 
9.55 
9.55 
9.55 
9.55 


514 
554 
593 
633 
673 


9.55 
9.55 
9.55 
9.55 
9.55 


712 

752 
791 
831 
870 


9.55 
9.55 
9.55 
9.56 
9.56 


910 
949 
989 
028 
067 


9.56  107 


41 
41 
41 
40 
41 

41 
40 

4' 
41 
40 

41 
40 
41 
40 
41 

40 
41 
40 
40 
41 

40 
40 
41 
40 
40 

40 
40 
40 
40 
40 

40 
40 
40 
40 
40 

40 
40 
39 
40 
40 

40 
39 
40 
40 
39 
40 
39 
40 
39 
40 

39 
40 
39 
39 
40 


0.46  303 
0.46  262 
0.46  221 
0.46  180 
0.46  139 


9.97  567 

9.97  563 

9.97  558 

9.97  554 

9.97  550 


0.46  098 
0.46  057 
0.46  016 
0.45  975 
0.45  935 


0.45 
0.45 
0.45 
0.45 
0.45 


894 
853 
813 

772 
731 


9.97  545 
9.97  541 
9.97  536 
9.97  532 
9.97  528 
9.97  523 
9.97  519 
9.97  515 
9.97  510 
9.97  506 


0.45 
0.45 
0.45 
0.45 
0.45 


691 
650 
610 
569 

529 


9.97  501 
9.97  497 
9.97  492 
9.97  488 
9.97  484 


0.45 
0.45 
0.45 
0,45 
0.45 


488 
448 
407 
367 
327 


9.97  479 
9.97  475 
9.97  470 
9.97  466 
9.97  461 


0.45 
0.45 
0.45 
0.45 
0.45 


286 
246 
206 
165 
125 


9.97  457 
9.97  453 
9.97  448 
9.97  444 
9.97  439 


885 
845 
805 
765 

725 


9.97  435 
9.97  430 
9.97  426 
9.97  421 
9.97  417 
9.97  412 
9.97  408 
9.97  403 
9.97  399 
9.97  394 


0.44  685 
0.44  645 


9.97  390 
9.97  385 


0.44 
0.44 
0.44 


605 
566 
526 


9.97 
9.97 


381 
376 


9.97  372 


0.44  486 
0.44  446 


0.44 
0.44 
0.44 


407 
367 
327 


9.97  367 

9.97  363 

9.97  358 

9.97  353 

9.97  349 


0.44 
0.44 
0.44 
0.44 
0.44 


288 
248 
209 
169 
130 


9.97  344 
9.97  340 
9.97  335 
9.97  331 
9.97  326 


0.44  090 
0.44  051 
0.44  Oil 
0.43  972 
0.43  933 


9.97  322 
9.97  317 
9.97  312 
9.97  308 
9.97  303 


0.43  893 


9.97  299 


11 

4.1 

40 

4.0 

4.8 

4.7 

5.5 

5-3 

6.2 

6.0 

6.8 

6.7 

13-7 

13-3 

20.5 

20.0 1 

27.3 

26.7 

34.2 

33-3 

3-9 
4.6 
5-2 
5-9 
6.5 
13.0 
19-5 
26.0 
32.5 


37  86  85 


3.7 

4.3 

4.9 

5.6 

6.2 
12.3 
18.S 
24.7124.0 
30.8130.0 


3.6 
4.2 
4.8 
5-4 
6.0 
12.0 
18.0 


3-5 

4.1 

4-7 

5-3 

5.8 

II. 7 

17.5 

23.3 

29.2 


// 

84 

5 

4 

6 

3-4 

o.S 

0.4 

7 

4.0 

0.6 

0.5 

8 

4-5 

0.7 

0.5 

0 

S.I 

0.8 

0.6 

10 

5-7 

0.8 

0.7 

20 

1 1.3 

1-7 

1.3 

30 

17.0 

2.5 

2.0 

40 

22.7 

3.3 

2.7 

SO 

28.3 

4.2 

3-3 

log  COS      d.      log  cot   c.  d.    log  tan      log  sin 
*i6o°         250°        *340°  70*^ 


d. 


Prop.  Pts. 


48 


Logarithms  of  thk  Trigonometric  Functions 


20^ 


^290" 


10 

11 
12 

13 
JL4 
l5 
16 
17 
18 

J^ 
20 

21 
22 
23 
2^ 
25 
26 
27 
28 

30 

31 
32 
33 

11 
3S 
36 
37 
38 

11 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


Io|?  sin 


9.53  405 
9.53  440 
9.53  475 
9.53  509 
9.53   544 


9.53 
9.. 53 
9.53 
9.53 
9.53 


578 
613 
647 
682 
716 


9.53 
9.53 
9.53 
9.53 
9.53 


751 

785 
819 
854 
888 


9.53  922 
9.53  957 

9.53  991 

9.54  025 
9.54  059 


(1. 


54  093 
54  127 
54  161 
54  195 
54  229 


54  263 
54  297 
54  331 
54  365 
54  399 


9.-54  '^33 

9.54  466 

9.54  500 

9.54  534 

9.54  567 


9.54  601 
9.54  635 
9.54  668 
9.54  702 
9.54  735 


9.54 
9.54 
9.54 
9.54 
9.54 


769 
802 
836 
869 
903 


9.54  936 

9.54  969 

9.55  003 
9.55  036 
9.55  069 


9.55 
9.55 
9.55 
9.55 
9.55 


102 
136 
169 

202 
235 


log  cos 


'159^ 


35 
35 
34 
35 
34 

35 
34 
35 
34 
35 

34 
34 
35 
34 
34 

35 
34 
34 
34 

34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

33 
34 
34 
33 
34 

34 
33 
34 
33 
34 

33 
34 
33 
34 
33 

33 
34 
33 
33 
33 

34 
33 
33 
33 
33 

33 
33 
33 
33 
33 

T" 

249^ 


log  tan    c.  (1. 


9.56 
9.56 
9.56 
9.56 
9.56 


107 
146 
185 
224 
264 


9.56 
9.56 
9.  .56 
9.56 
9.56 


303 
342 
381 
420 
459 


9.56 
9.56 
9.56 
9.56 
9.56 


9.56 
9.56 
9.56 
9.56 
9.56 


498 
537 
576 
615 
654 
693" 
732 
771 
810 
849 


9.56 
9.56 
9.56 
9.57 
9.57 


887 
926 
965 
004 
042 


9.57 
9.57 
9.57 
9.57 
9.57 


081 
120 
158 
197 
235 


9.57 
9.57 
9.57 
9.57 
9.57 


274 
312 

351 
389 
428 


9.57 
9.57 
9.57 
9.57 
9.57 


466 
504 
543 
581 
619 


9.57 
9.57 
9.57 
9.57 
9.57 


658 
696 

734 
772 
810 


9.57 
9.57 
9.57 
9.57 
9.58 


849 
887 
925 
963 
001 


9.58 
9.58 
9.58 
9.58 
9.58 


039 
077 
115 
153 
191 


9.58 
9.58 
9.58 
9.58 
9.58 


229 
267 
304 
342 
380 


9.58  418 
log  cot 


39 
39 
39 

40 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
38 

39 
39 
39 
38 
39 


39 
38 
39 

38 
39 
38 
39 
38 

38 
39 
38 
38 
39 

38 
38 
38 
38 
39 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
37 
38 
38 
38 


log  cot   log  COS  j  d. 


0.43  893 

0.43  854 

0.43  815 

0.43  776 

0.43  736 


9.97  299 
9.97  294 
9.97  289 
9.97  285 
9.97  280 


0.43  697 

0.43  658 

0.43  619 

0.43  580 

0.43  541 


9.97  276 
9.97  271 
9.97  266 
9.97  262 
9.97  257 


0.43  502 
0.43  463 
0.43  424 
0.43  385 
0.43  346 


0.43  307 
0.43  268 


0.43 
0.43 
0.43 


229 
190 
151 


0.43  113 
0.43  074 
0.43  035 
0.42  996 
0.42  958 


0.42  919 
0.42  880 
0.42  842 
0.42  803 
0.42  765 


0.42  726 
0.42  688 
0.42  649 
0.42  611 
0.42  572 


0.42  534 
0.42  496 
0.42  457 
0.42  419 
0.42  381 


9.97  252 
9.97  248 
9.97  243 
9.97  238 
9.97  234 


9.97  229 
9.97  224 
9.97  220 
9.97  215 
9.97  210 


9.97 
9.97 
9.97 
9.97 
9.97 


206 
201 
196 
192 
187 


9.97 
9.97 
9.97 
9.97 
9.97 
9.97 
9.97 
9.97 
9.97 
9.97 


182 
178 
173 
168 
163 
159 
154 
149 
145 
140 


0.42  342 
0.42  304 


0.42 
0.42 
0.42 


266 
228 
190 


0.42  151 
0.42  113 
0.42  075 
0.42  037 
0.41  999 


0.41 
0.41 
0.41 
0.41 
0.41 


961 
923 

885 
847 
809 


9.97 
9.97 
9.97 
9.97 
9.97 


135 
130 
126 
121 
116 


9.97  111 
9.97  107 
9.97  102 
9.97  097 
9.97  092 


9.97  087 
9.97  083 
9.97  078 
9.97  073 
9.97  068 


0.41 
0.41 
0.41 
0.41 
0.41 


771 
733 
696 
658 
620 


9.97  063 
9.97  059 
9.97  054 
9.97  049 
9.97  044 


'^339^ 


0.41   582 
log  tan 

69^ 


9.97  039 
9.97  035 
9.97  030 
9.97  025 
9.97  020 
9.97  015 


log  sin  I   d. 


Prop.  Pts. 


40  I  39 


4.0  3.9 

4.7  4.6 

5.3!  5.2 
6.0;  s.g 
6.7!  6.S 
13-3  13.0 
30  20.0  ig.s 
40  26.7,26.0 
So!33.3  32.S 


"  38    37 


3.8 

4.4 
5-1 
5.7 
6.3 
12.7 
19.0 
25.3 


3-7 
4-3 
4.9 
5.6 
6.2 
12.3 
18.S 
24.7 
30.8 


/' 

35 

6 

35 

7 

4.1 

8 

4.7 

9 

5.3 

10 

5.8 

20 

II. 7 

30 

17.5 

40 

23.3 

SO 

29.2 

30 


3-4 
4.0 
4-5 
S-i 
5-7 
11.3 
17.0 
22.7 
28.3 


5    4 


3.3:0.5 
3.80.6 
4.40.7 
5.0  0.8 
5-5  0.8 
ii.o  1.7 

16.S  2.S 

22.0^3.3 
50I27.SI4.2 


0.4 

O.S 

0.5 
0.6 

0.7 

1-3 
2.0 
2.7 
3.3 


Prop.  Pts. 


Logarithms  of  the  Trigonometric  Functjons 


2r 


^29^ 


10 

11 
12 
13 
14 
15 
16 
17 
18 

}1 
20 

21 

22 
23 
U_ 
25 
26 
27 
28 
29 
30 
31 
32 
33 

"35" 
36 
37 
38 
39_ 

40 

41 
42 
43 

ii 

45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 

11 
60 


log  sill 

9.55 
9.55 
9.55 
9.55 
9.55 


433 
466 
499 
532 
564 
597 
630 
663 
695 
728 
761 
793 
826 
858 
891 
923 
956 
988 
021 
053 
085 
118 
150 
182 
215 
247 
279 
311 
343 
375 
408 
440 
472 
504 
536 
568 
599 
631 
663 
695 
727 
759 
790 
822 
854 
886 
917 
949 
980 
012 
044 
075 
107 
138 
169 
201 
232 
264 
295 
326 
9.57  358 
log  COS 


9.55 
9.55 
9.55 
9.55 
9.55 
9.55 
9.55 
9.55 
9.55 
9.55 
9.55 
9.55 
9.55 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.  .56 
^9^56" 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.  .56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.57 
9.57 
9.57 
9.57 
9.57 
9.57 
9.57 
9.57 
9.57 
9.57 
9.57 


d. 


33 
33 
33 
32 
33 

33 
33 
32 
33 
33 

32 
33 
32 
33 
32 

33 
32 
33 
32 
32 

33 
32 
32 
33 
32 

32 
32 
32 
32 
33 

32 
32 
32 
32 
32 

31 
32 
32 
32 
32 

32 

31 
32 
32 
32 

31 
32 
31 
32 
32 

31 
32 
31 
31 
32 

31 
32 
31 
31 
32 


log  tan  c.  d.    log  cot      log  cos 


9.58 
9.58 
9.58 
9.  .58 
9.58 


418 
455 
493 
531 
569 


9.58 
9.58 
9.58 
9.58 
9.58 


606 
644 

681 
719 

757 


9.58 
9.58 
9.58 
9.58 
9.58 


794 
832 
869 
907 
944 


9.58 
9.59 
9.59 
9.59 
9.59 


98  i 
019 
056 
094 
131 


9.59 
9.59 
9.59 
9.59 
9.59 


168 
205 
243 
280 
317 


9.59 
9.59 
9.59 
9.59 
9.  .59 


354 
391 
429 
466 
503 


9.59 
9.59 
9.59 
9.59 
9.59 


540 
577 
614 
651 
688 


9.  .59 
9.59 
9.59 
9.59 
9.59 


725 
762 
799 
835 
872 


9.59 
9.59 
9.  .59 
9.60 
9.60 


9.60 
9.60 
9.60 
9.60 
9.60 


909 
946- 
983 
019 
056 
093 
130 
166 
203 
240 


9.60 
9.60 
9.60 
9.60 
9.60 


276 
313 
349 

386 
422 


9.60 
9.60 
9.60 
9.60 
9.60 


459 
495 
532 
568 
605 


*i58= 


9.60  641 
log  cot 

*338° 


37 
38 
38 
38 
37 

38 
37 
38 
38 

37 

38 
37 
38 
37 
37 

38 
37 
38 
37 
37 

37 
38 
37 
37 
37 

37 
38 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
36 
37 
37 

37 
37 
36 
37 
37 

37' 
36 
37 
37 
36 

37 
36 
37 
36 
37 

36 
37 
36 
37 
36 


0.41 
0.41 
0.41 
0.41 
0.41 


582 
545 
507 
469 
431 


0.41 
0.41 
0.41 
0.41 
0.41 


394 
356 
319 
281 
243 


0.41 
0.41 
0.41 
0.41 
0.41 


206 
168 
131 
093 
056 


0.41  019 
0.40  981 
0.40  944 
0.40  906 
0.40  869 


0.40  832 
0.40  795 
0.40  757 
0.40  720 
0.40  683 


0.40  646 
0.40  609 
0.40  571 
0.40  534 
0.40  497 


0.40  460 
0.40  423 
0.40  386 
0.40  349 
0.40  312 


0.40  275 
0.40  238 
0.40  201 
0.40  165 
0.40  128 


0.40  091 
0.40  054 
0.40  017 
0.39  981 
0.39  944 


0.39  907 
0.39  870 
0.39  834 
0.39  797 
0.39  760 


0.39  724 
0.39  687 
0.39  651 
0.39  614 
0.39  578 


0.39  .541 
0.39  505 
0.39  468 
0.39  432 
0.39  395 
0.39  359 

log  tan 

68^ 


9.97  015 
9.97  010 
9.97  005 
9.97  001 
9.96  996 


9.96  991 
9.96  986 
9.96  981 
9.96  976 
9.96  971 


9.96  966 
9.96  962 
9.96  957 
9.96  952 
9.96  947 


9.96  942 
9.96  937 
9.96  932 
9.96  927 
9.96  922 


9.96  917 
9.96  912 
9.96  907 
9.96  903 
9.96  898 


9.96  893 
9.96  888 
9.96  883 
9.96  878 
9^96  873 


9.96  868 
9.96  863 
9.96  858 
9.96  853 
9.96  848 


9.96  843 
9.96  838 
9.96  833 
9.96  828 
9.96  823 


9.96  818 
9.96  813 
9.96  808 
9.96  803 
9.96  798 


9.96  793 
9.96  788 
9.96  783 
9.96  778 
9.96  772 


9.96  767 
9.96  762 
9.96  757 
Q.96  752 
9.96  747 


9.96  742 
9.96  737 
9.96  732 
9.96  727 
9.96  722 
9.96  717 
log  sin 


d. 


5 

5 
4 
S 
5 

5 
S 
S 

S 

s 

4 
5 
5 
5 
S 

S 

5 
5 
5 
5 

S 
5 
4 
5 
5 

S 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
S 

S 

s 
s 

s 
s 
s 

5 

s 

5 
S 
5 
6 
5 

5 
S 
5 
S 
5 

5 

S 
5 
S 

S 

T" 


Prop.  Pts. 


" 

88 

87 

6 

3.8 

3.7 

7 

4.4 

4.3 

8 

S.I 

4.g 

0 

^■7 

,s.o 

10 

6.3 

6.2 

20 

12.7 

12.3 

30 

19.0 

18.S 

40 

2.S.3 

24.7 

SO 

31.7 

30.8 

3.6 

4.2 

4.8 

S.4 
6.0 
12.0 
18.0 
24.0 
30.0 


"  88  82  81 


3.3 
3.9 

4.4 
S.o 
5-5 
li.o 
i6.s 


3.2]  3.1 
3.7|  3.6 
4.3'  4-1 
4.8  4.6 
5.3  5.2 
10.7  10.3 
16.0  15.5 
22.0J21.3  20.7 
27.5126.7:25.8 


6  6  I  4 

0.6  0.5  0.4 
0.7  0.6  o.j 
0.8  0.7  o.s 
0.9 

I.O 

2.0 

3.0 
4.0 

S-o 


Prop.  Pts. 


Logarithms  of  the  Teigonometbic  Fuxctions 


22° 

*\\'P 

202''            *292° 

0 

log  sill 

1   d. 

log  tan 

c.  d. 

log  cot 

log  cos 

\  d. 

Prop.  Pts. 

9.57  358 

1 

9.60  641 

36 
37 
36 
36 

0.39  359 

9.96  717 

6 

60 

1 

9.57  389 

9.60  677 

0.39  323 

9.96  711 

59 

2 

9.57  420 

31 

9.60  714 

0.39  286 

9.96  706 

5 
5 
5 

58 

3 

9.57  451 

9.60  750 

0.39  250 

9.96  701 

57 

4 

9.57  482 

32 

9.60  786 

37 

36 
36 
36 
36 
37 

36 
36 
36 
36 
36 

0.39  214 

9.96  696 

56 

55 

5 

9.57  514 

9.60  823 

0.39  177 

9.96  691 

6 

9.57  545 

9.60  859 

0.39  141 

9.96  686 

5 

54 

7 

9.57  576 

31 
31 
31 

9.60  895 

0.39  105 

9.96  681 

53 

8 

9.57  607 

9.60  931 

0.39  069 

9.96  676 

6 

52 

9 
10 

9.57  638 

9.60  967 

0.39  033 

9.96  670 

5 

51 
50 

" 

37  1  SA 

36 

9.57  669 

9.61  004 

0.38  996 

9.96  665 

11 

9.57  700 

31 

9.61  040 

0.38  960 

9.96  660 

5 

49 

6 

3.7 

3.6 

3-5 

12 

9.57  731 

9.61  076 

0.38  924 

9.96  655 

48 

7 
8 

4.3 
4.9 

■;.6 

4.2 
4.8 

';.4 

4 
4 

5 

I 
7 
3 

13 

9.57  762 

31 
31 

9.61   112 

0.38  888 

9.96  650 

5 
5 

47 

9 

14 

9.57  793 

9.61   148 

0.38  852 

9.96  645 

46 

10 

6.2 
12.3 
18.5 

6.0 

5 

8 

7 
5 

15 

9.57  824 

9.61   184 

36 
36 
36 
36 
36 

36 
36 
36 
36 
36 

0.38  816 

9.96  640 

6 

45 

30 

18.0 

17 

16 

9.57  855 

31 

9.61   220 

0.38  780 

9.96  634 

44 

40 

24.7 

24.0 

23 

3 

17 

9.57  885 

30 

9.61  256 

0  38  744 

9.96  629 

5 

43 

SO 

30.8  30.0 

29.2           1 

18 

9.57  916 

31 

9.61   292 

0.38  708 

9.96  624 

5 
S 
5 

6 

42 

19 

9.57  947 

31 

9.61  328 

0.38  672 

9.96  619 

41 
40 

20 

9.57  978 

9.61  364 

0.38  636 

9.96  614 

21 

9.58  008 

30 

9.61  400 

0.38  600 

9.96  608 

39 

22 

9.58  039 

31 

9.61  436 

0.38  564 

9.96  603 

5 

38 

23 

9.58  070 

31 

9.61  472 

0.38  528 

9.96  598 

5 
5 
5 

6 

37 

24 

9.58  101 

30 

9.61  508 

0.38  492 

9.96  593 

36 
35 

25 

9.58   131 

9.61   544 

0.38  456 

9.96  588 

7.f^ 

9.58  162 

31 

9.61  579 

35 

0.38  421 

9.96  582 

34 

27 

9.58  192 

30 

9.61  615 

36 
36 
36 

0.38  385 

9.96  577 

5 

Zl 

»2 

31 

3U 

28 

9.58  223 

31 

9.61  651 

0.38  349 

9.96  572 

5 

32 

6 

3-2 

3.1 

3.0 

?,9 

9.58  253 

9.61  687 

0.38  313 

9.96  567 

31 

7 

3-7 

3.6 

3.5 

30 

31 

35 
36 

5 
6 

30 

8 
9 

4.3 
4.8 

4.1 
4.6 

4.0 
4.5 

9.58  284 

9.61   722 

0.38  278 

9.96  562 

31 

9.58  314 

30 

9.61  758 

0.38  242 

9.96  556 

29 

10 

5-3 

5-2 

S.o 

32 

9.58  345 

31 

9.61   794 

36 
36 

0.38  206 

9.96  551 

5 

28 

20 

10.7 

10.3 

lO.O 

33 

9.58  375 

9.61  830 

0.38  170 

9.96  546 

27 

40 

21. •? 

20.7 

20.0 

34 

9.58  406 

30 

9.61  865 

36 

0.38  135 

9.96  541 

6 

26 

50 

26.7 

2S.8 

25.0 

35 

9.58  436 

9.61  901 

0.38  099 

9.96  535 

25 

36 

9.58  467 

31 

9.61  936 

35 

0.38  064 

9.96  530 

5 

24 

37 

9.58  497 

30 

9.61  972 

36 
36 

0.38  028 

9.96  525 

5 

23 

38 

9.58  527 

9.62  008 

0.37  992 

9.96  520 

6 
5 

22 

39 

9.58  557 

31 

9.62  043 
9.62  079 

36 

0.37  957 

9.96  514 

21 
20 

40 

9.58  588 

0.37  921 

9.96  509 

41 

9.58  618 

30 

9.62  114 

35 

0.37  886 

9.96  504 

S 

19 

42 

9.58  648 

30 

9.62  150 

36 

0.37  8.50 

9.96  498 

18 

43 

9.58  678 

30 

9.62  185 

35 
36 

0.37  815 

9.96  493 

5 

17 

44 

9.58  709 

31 

9.62  221 

0.37  779 

9.96  488 

5 

16 

zv    0 

a 

45 

9.58  739 

9.62  256 

0.37  744 

9.96  483 

15 

6 

2.90.6 

0.5 

46 

9.58   769 

30 

9.62  292 

36 

0.37  708 

9.96  477 

6 

14 

7 
g 

3.40.7 

3  nl"  81 

0.6 
0.7 
0.8 

47 

9.58   799 

30 

9.62  327 

35 

0.37  673 

9.96  472 

5 

13 

g 

4.4 

0.0 

48 

9.58  829 

30 

9.62  362 

35 
36 

0.37  638 

9.96  467 

6 

12 

10 

4.8 

I.O 

0.8 

49 

9.58  859 

30 

9.62  398 

0.37  602 

9.96  461 

11 

20 

9-7 

2.0 

1-7 

50 

9.58  889 

9.62  433 

0.37  567 

9.96  456 

10 

40 

19-3 

4.0 

3.3 

51 

9.58  919 

30 

9.62  468 

35 

0.37  532 

9.96  451 

5 

9 

so 

24.2 

S.o 

4.2 

52 

9.58  949 

30 

9.62  504 

36 

0.37  496 

9.96  445 

6 

8 

53 

9.58  979 

30 

9.62  539 

35 

0.37  461 

9.96  440 

S 

7 

54 
55 

9.59  009 

30 
30 

9.62  574 

35 
35 

0.37  426 

9.96  435 

6 

6 

5 

9.59  039 

9.62  609 

0.37  391 

9.96  429 

56 

9.59  069 

30 

9.62  645 

3(J 

0.37  355 

9.96  424 

5 

4 

57 

9.59  098 

29 

9.62  680 

35 

0.37  320 

9.96  419 

5 
6 

3 

58 

9.59  128 

30 

9.62   715 

35 

0.37  285 

9.96  413 

2 

59 
60 

9.59  158 

30 
30 

9.62  750 

35 
35 

0.37  250 

9.96  408 

5 

1 
0 

9.59  188 

9.62  785 

0.37  215 

9.96  403 

log  COS 

d. 

log  cot 

eTT 

log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*i57° 

247° 

*337° 

67° 

Logarithms  of  the  Trigonometric  Functions 


23° 

* 

113°        203°        *293°        1 

log  sin 

d. 

log  tan 

c.  d. 

log  cot 

log  cos 

d. 

Prop.  P(s. 

9.59  188 

9.62  785 

35 
35 

0.37  215 

9.96  403 

5 

60 

1 

9.59  218 

29 

3° 

9.62  820 

0.37   180 

9.96  397 

5 
5 

6 

59 

2 

9.59  247 

9.62  855 

0.37   145 

9.96  392 

58 

3 

9.59  277 

9.62  890 

36 

0.37   110 

9.96  387 

57 

4 

5 

9.59  307 

29 

9.62  926 

35 

0.37  074 

9.96  381 

S 
6 
5 
5 
6 

56 

55 

9.59  336 

9.62  961 

0.37  039 

9.96  376 

6 

9.59  366 

30 

9.62  996 

35 
35 

0.37  004 

9.96  370 

54 

7 

9.59  396 

29 
30 
29 

9.63  031 

0.36  969 

9.96  365 

53 

8 

9.59  425 

9.63  066 

0.36  934 

9.96  360 

52 

9 

9.59  455 

9.63   101 

34 

0.36  899 

9.96  354 

5 

51 

„ 

ifi 

HFi 

34 

10 

9.59  484 

9.63   135 

0.36  865 

9.96  349 

6 

50 

11 

9.59  514 

30 

9.63   170 

35 

0.36  830 

9.96  343 

49 

6 

3.6 

3.5 

3.4 

12 

9.59  543 

29 
30 
29 
30 

9.63  205 

35 
35 
35 
35 

0.36  795 

9.96  338 

5 
6 

48 

7 
8 

4.2 
4.8 

5-4 

4.1 

4-7 

S.2 

4.0 

4.5 
5-1 

13 

9.59  573 

9.63   240 

0.36  760 

9.96  333 

47 

9 

14 

9.59  602 

9.63   275 

0.36  725 

9.96  327 

5 

46 

10 

6.0 

5.« 
11.7 

17.5 

5.7 
11.3 
17.0 

15 

9.59  632 

9.63   310 

0.36  690 

9.96  322 

6 

45 

30 

18.0 

16 

9.59  661 

9.63   345 

0.36  655 

9.96  316 

44 

40 

24.0 

23.3 

22.7 

17 

9.59  690 

9.63  379 

0.36  621 

9.96  311 

6 

43 

SO 

30.0 

29.2 

28.3 

18 

9.59  720 

29 
29 

9.63  414 

35 
35 

0.36  586 

9.96  305 

5 
6 

42 

19 
20 

9.59  749 

9.63  449 

0.36  551 

9.96  300 

41 
40 

9.59  778 

9.63   484 

0.36  516 

9.96  294 

21 

9.59  808 

30 

9.63  519 

35 

0.36  481 

9.96  289 

39 

22 

9.59  837 

9.63  553 

0.36  447 

9.96  284 

6 

5 
6 

38 

23 

9.59  866 

9.63  588 

35 
34 

0.36  412 

9.96  278 

37 

24 

25 

9.59  895 

29 

9.63  623 

0.36  377 

9.96  273 

36 

35 

9.59  924 

9.63  657 

0.36  343 

9.96  267 

26 
27 

9.59  954 
9.59  983 

30 
29 

9.63  692 
9.63   726 

35 

34 

0.36  308 
0.36  274 

9.96  262 
9.96  256 

5 
6 

34 
33 

ff 

80 

29 

28 

28 

9.60  012 

29 

9.63   761 

35 

0.36  239 

9.96  251 

32 

6 

3.0 

2.9 

2.8 

29 

9.60  041 

29 

9.63  796 

0.36  204 

9.96  245 

31 

7 

3.5 

3.4 

3-3 

4.0 

4.S 

3.9 

4.4 

3.7 
4.2 

30 

9.60  070 

9.63  830 

0.36  170 

9.96  240 

30 

9 

31 

9.60  099 

29 

9.63  865 

35 

0.36  135 

9.96  234 

29 

10 

S.o 

4.8 

4.7 

32 

9.60  128 

29 

9.63  899 

34 

0.36  101 

9.96  229 

5 
6 

28 

20 
30 
40 

lO.O 

15.0 

20.0 

9-7 
14.5 
10.3 

9-3 
14.0 
18.7 

33 

9.60  157 

9.63  934 

0.36  066 

9.96  223 

27 

34 

9.60  186 

29 

9.63  968 

35 

0.36  032 

9.96  218 

6 

26 

25 

SO 

25.0 

24.2 

23.3 

35 

9.60  215 

9.64  003 

0.35  997 

9.96  212 

36 

9.60  244 

29 

9.64  037 

34 

0.35  963 

9.96  207 

S 

24 

37 

9.60  273 

29 

9.64  072 

3  J 

0.35  928 

9.96  201 

6 

23 

38 

9.60  302 

29 

9.64   106 

0.35  894 

9.96  196 

S 
6 
5 

22 

39 

9.60  331 

28 

9.64  140 

35 

0.35   860 

9.96  190 

21 
20 

40 

9.60  359 

9.64   175 

0.35  825 

9.96  185 

41 

9.60  388 

29 

9.64  209 

34 

0.35   791 

9.96  179 

6 

19 

42 

9.60  417 

29 

9.64  243 

34 

0.35  757 

9.96  174 

5 
6 
6 
5 

18 

43 

9.60  446 

29 
28 
29 

9.64  278 

35 

0.35   722 

9.96  168 

17 

44 

9.60  474 

9.64  312 

34 

0.35  688 

9.96  162 

16 
15" 

"   f 

60 

S     5 
60.S 

45 

9.60  503 

9.64  346 

0.35  654 

9.96  157 

46 

9.60  532 

29 

9.64  381 

35 

0.35  619 

9.96  151 

6 

14 

70 
80 
9  0 

7  0.6 
80.7 
90.8 

47 

9.60  561 

29 

9.64  41i 

34 

0.35   585 

9.96  146 

5 

13 

48 

9.60  589 

9.64  449 

34 

0.35   551 

9.96  140 

12 

10  I 

00.8 

49 

9.60  618 

29 

9.64  483 

34 

0.35  517 

9.96  135 

5 

11 

20  2 

0  1.7 

50 

9.60  646 

9.64  517 

0.35  483 

9.96  129 

10 

404 

03.3 

51 

9.60  675 

29 

9.64  552 

35 

0.35  448 

9.96  123 

6 

9 

SOS 

04.2 

52 

9.60  704 

29 

9.64  586 

34 

0.35  414 

9.96  118 

5 

8 

53 

9.60  732 

9.64  620 

34 

0.35  380 

9.96  112 

7 

54 

55 

9.60  761 

29 
28 

9.64  654 

34 
34 

0.35  346 

9.96  107 

5 
6 

6 

5 

9.60  789 

9.64  688 

0.35  312 

9.96  101 

56 

9.60  818 

29 

9.64   722 

34 

0.35  278 

9.96  095 

6 

4 

57 

9.60  846 

28 

9.64   756 

34 

0.35  244 

9.96  090 

5 

3 

58 

9.60  875 

?9 

9.64  790 

34 

0.35  210 

9.96  084 

6 

2 

59 
60 

9.60  903 

28 

9.64  824 

34 

34 

0.35  176 

9.96  079 

5 
6 

1 
0 

9.60  931 

9.64  858 

0.35   142 

9.96  073 

log  COS 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

1 

Prop.  Pts. 

*I56°        2 

46^ 

*336" 

66° 

1 

R9. 


Logarithms  of  the  Trigoxometric  Functions 


24° 


ni4^ 


204^^ 


^294" 


log  sin      d. 


log  tan   c.  d.    log  cot      log  cos 


d. 


Prop.  Pts. 


0 

1 
2 
3 

5 

6 

7 

8 

9_ 

10 

11 

12 

13 

14 

15 

16 

17 

18 

i2. 
20 

21 
22 
23 

25 
26 

27 
28 

30 

51 
32 
33 

ii 
35 
36 
37 
38 

40 

41 
42 
43 

ii 

45 
46 
47 
48 
49 
50 
51 
52 
53 

it 

55 
56 

5i 

58 

ii 
60 


9.60 
9.60 
9.60 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9-62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62   595 


931 

960 

988 

016 

045 

073 

101 

129 

158 

186 

214 

242 

270 

298 

326 

354 

382 

411 

438 

466 

494  j 

522 

550 

578  : 

606  i 

634  i 

662 

689 

717 

745 

773 

800 

828 

856 

883 

911 

939 

966 

994  i 

021  j 

049 

076  ! 

104  I 

131  I 

159  I 

186  : 

214 

241  , 

268  I 

296 

323  i 

350  I 

377  I 

405  1 

432 

459 

486 

513 

541 

568 


29 
28 
28 
29 
28 

28 
28 
29 
28 
28 

28 
28 
28 
28 
28 

28 
29 
27 
28 
28 

28 
28 
28 
28 
28 

28 
27 
28 
28 
28 

27 
28 
28 
27 
28 

28 
27 
28 
27 
28 

27 
28 
27 
28 
27 
28 
27 
27 
28 
27 

27 
27 
28 
27 
27 

27 
27 
28 
27 
27 


log  cos   I   d. 


9.64 
9.64 
9.64 
9.64 
9.64 


858 
892 
926 
960 
994 


9.65 
9.65 
9.65 
9.65 
9.65 


028 
062 
096 
130 
164 


9.65 
9.65 
9.65 
9.65 
9.65 


197 
231 
265 
299 
333 


9.65 
9.65 
9.65 
9.65 
9.65 


366 
400 
434 
467 
501 


9.65 
9.65 
9.65 
9.65 
9.65 


535 
568 
602 
636 
669 


9.65 
9.65 
9.65 
9.65 
9.65 


703 
736 
770 
803 
837 


9.65 
9.65 
9.65 
9.65 
9.66 


870 
904 
937 
971 
004 


9.66 
9.66 
9.66 
9.66 
9.66 


038 
071 
104 
138 
171 


9.66 
9.66 
9.66 
9.66 
9.66 
9.66 
9.66 
9.66 
9.66 
9.66 


204 
238 
271 
304 
337 


371 
404 
437 
470 
503 


9.66 
9.66 
9.66 
9.66 
9.66 


537 
570 
603 
636 
669 


9.66 
9.66 
9.66 
9.66 
9.66 


702 
735 
768 
801 
834 


^155° 


245 


9.66  867 
log  cot 


•^335° 


34 
34 
34 
34 
34 

34 
34 
34 
34 
33 

34 
34 
34 
34 
33 

34 
34 
33 
34 
34 

3i 
34 
34 
33 
34 

35 
34 
33 
34 
33 

34 
33 
34 
33 
34 

33 
33 
34 
33 
33 

34 
33 
33 
33 
34 

33 
33 
33 
33 
34 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

~ 


0.35  142 
0.35  108 
0.35  074 
0.35  040 
0.35  006 


9.96  073 
9.96  067 
9.96  062 
9.96  056 
9.96  050 


0.34  972 
0.34  938 
0.34  904 
0.34  870 
0.34  836 


9.96  045 
9.96  039 
9.96  034 
9.96  028 
9.96  022 


0.34  803 

0.34  769 

0.34  735 

0.34  701 

0.34  667 


9.96  017 
9.96  Oil 
9.96  005 
9.96  000 
9.95  994 


0.34  634 

0.34  600 

0.34  566 

0.34  533 

0.34  499 


9.95  988 
9.95  982 
9.95  977 
9.95  971 
9.95  965 


0.34  465 
0.34  432 
0.34  398 
0.34  364 
0.34  331 


9.95  960 
9.95  954 
9.95  948 
9.95  942 
9.95  937 


0.34  297 
0.34  264 
0.34  230 
0.34  197 
0.34  163 


9.95  931 
9.95  925 
9.95  920 
9.95  914 
9.95  908 


0.34  130 
0.34  096 
0.34  063 
0.34  029 
0.33  996 


9.95  902 
9.95  897 
9.95  891 
9.95  88i 
9.95  879 


0.33  962 
0.33  929 
0.33  896 
0.33  862 
0.33  829 


9.95  873 
9.95  868 
9.95  862 
9.95  856 
9.95  850 


0.33 
0.33 
0.33 
0.33 


796 
762 
729 
696 


0.33  663 


9.95  844 
9.95  839 
9.95  833 
9.95  827 
9.95  821 


0.33  629 
0.33  596 


0.33 
0.33 


563 
530 


0.33  497 


9.95  815 

9.95  810 

9.95  804 

9.95  798 

9.95  792 


0.33  463 
0.33  430 


0.33 
0.33 


397 
364 


0.33  331 


9.95 
9.95 
9.95 
9.95 
9.95 


786 
780 
775 
769 
763 


0.33 
0.33 
0.33 
0.33 
0.33 
0.33 


298 
265 
232 
199 
166 
133 


log  tau 

~65^ 


9.95  757 

9.95  751 

9.95  745 

9.95  739 

9.95  733 

9.95  728 


log  sin  j   d. 


34    33 


3.4 
4.0 
4.5 

S.I 

5-7 
11.3 
30  17.0 
40  22.7 
sol  28.3 


3.3 
3.9 

4.4 
5.0 
S-5 

II.O 

16.S 
22.0 
27.S 


// 

29 

28 

6 

2.0 

2.8 

7 

3.4 

3-3 

8 

3.9 

3.7 

9 

4.4 

4.2 

10 

4.8 

4-7 

20 

9.7 

9-3 

30 

14.5 

14.0 

40 

19-3 

18.7 

SO 

24.2 

23.3 

27 

2.7 
3.2 
3.6 
4.1 
4.5 
9.0 
13.S 
18.0 
22. s 


6    5 


Prop.  Pts. 


53 


Logarithms  of  the  Trigonometric  Fuxctions 


25< 


^115'     205- 


'"295" 


log  sin 


d. 


log  tan 


c.  d. 


log  cot 


log  COS 


d. 


Prop.  Pts. 


5 
6 
7 
8 
9^ 

10 

11 
12 
13 
14 


20 

21 
22 
23 
2^ 

25 
26 
27 
28 
2^ 

30 

31 
32 
33 

ii 

35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


50 

51 

52 
53 

ii 
55 
56 
57 
58 
^ 
60 


9.62  595 
9.62  622 
9.62  649 
9.62  676 
9.62  703 


9.62  730 
9.62  757 
9.62  784 
9.62  81] 
9.62  838 


9.62  865 
9.62  892 
9.62  918 
9.62  945 
9.62  972 


9.62  999 

9.63  026 
9.63  052 
9.63  079 
9.63  106 


9.63 
9.63 
9.63 


133 
159 
186 


9.63  213 
9.63  239 


9.63  266 
9.63  292 
9.63  319 
9.63  345 
9.63  372 


9.63  398 
9.63  425 
9.63  451 
9.63  478 
9.63  504 


9.63  531 
9.63  557 
9.63  583 
9.63  610 
9.63  636 


63  662 

63  689 

63  715 

63  741 

63  767 


63  794 
63  820 
63  846 
63  872 
63  89S 


9.63  924 
9.63  950 

9.63  976 

9.64  002 
9.64  028 


9.64  054 
9.64  080 
9.64  106 
9.64  132 
9.64  158 
9.64  184 

log  cos 

*I54° 


27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
26 
27 
27 
27 

27 
26 
27 
27 
27 

26 
27 
27 
26 
27 

26 
27 
26 
27 
26 

27 
26 
27 
26 
27 

26 
26 
27 
26 
26 

27 
26 
26 
26 
27 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

T" 

244° 


9.66 
9.66 
9.66 
9.66 
9.66 


867 
900 
933 
966 
999 


9.67 
9.67 
9.67 
9.67 
9.67 


032 
065 
098 
131 
163 


9.67 
9.67 
9.67 
9.67 
9.67 


196 
229 
262 
295 
327 


9.67 
9.67 
9.67 
9.67 
9.67 


360 
393 
426 
458 
491 


9.67 
9.67 
9.67 
9.67 
9.67 


524 
556 
589 
622 
654 


9.67 
9.67 
9.67 
9.67 
9.67 


687 
719 

752 
785 
817 


9.67 
9.67 
9.67 
9.67 
9.67 


850 
882 
915 
947 
980 


9.68 
9.68 
9.68 
9.68 
9.68 


012 
044 
077 
109 
142 


9.68 
9.68 
9.68 
9.68 
9.68 


174 
206 
239 
271 
303 


9.68 
9.68 
9.68 
9.68 
9.68 


336 
368 
400 
432 
465 


9.68 
9.68 
9.68 
9.68 
9.68 


497 
529 
561 
593 
626 


9.68 
9.68 
9.68 
9.68 
9.68 


658 
690 
722 
754 
786 


33 
33 

33 
33 

33 
33 
33 
32 
33 

33 
33 
33 
32 
33 

33 
33 
32 
a 
33 

32 
a 
33 
32 
33 

32 
33 
33 
32 
33 

32 
33 
32 
33 
32 

32 
33 
32 
33 
32 

32 
33 
32 
32 
33 

32 
32 
32 
33 
32 

32 
32 
32 
33 
32 

32 
32 
32 
32 
32 


9.68  818 


log  cot    c.  d 


^334" 


0.33  133 

0.33  100 

0.33  067 

0.33  034 

0.33  001 


9.95 
9.95 
9.95 
9.95 
9.95 


728 
722 
716 
710 
704 


0.32  968 
0.32  935 
0.32  902 
0.32  869 
0.32  837 


9.95  698 
9.95  692 
9.95  686 
9.95  680 
9.95  674 


0.32 
0.32 
0.32 
0.32 


804 
771 
738 
705 


0.32  673 


9.95  668 
9.95  663 
9.95  657 
9.95  651 
9.95  645 


0.32  640 

0.32  607 

0.32  574 

0.32  542 

0.32  509 


9.95  639 
9.95  633 
9.95  627 
9.95  621 
9.95  615 


0.32  476 
0.32  444 
0.32  411 
0.32  378 
0.32  346 


9.95 
9.95 
9.95 
9.95 
9.95 


609 
603 
597 
591 

585 


0.32  313 
0.32  281 
0.32  248 
0.32  215 
0.32  183 


9.95  579 

9.95  573 

9.95  567 

9.95  561 

9.95  555 


0.32  150 
0.32  118 
0.32  085 
0.32  053 
0.32  020 


9.95 
9.95 
9.95 
9.95 
9.95 


549 
543 

537 
531 

525 


0.31 
0.31 
0.31 
0.31 
0.31 


988 
956 
923 
891 

858 


9.95  519 
9.95  513 


9.95 
9.95 


507 
500 


9.95  494 


0.31 
0.31 
0.31 
0.31 
0.31 
0.31 
0.31 
0.31 
0.31 
0.31 


826 
794 
761 
729 
697 
664 
632 
600 
568 
535 


9.95  488 

9.95  482 

9.95  476 

9.95  470 

9.95  464 


9.95  458 
9.95  452 
9.95  446 
9.95  440 
9.95  434 


0.31  503 

0.31  471 

0.31  439 

0.31  407 

0.31  374 


9.95  427 
9.95  421 
9.95  415 
9.95  409 
9.95  403 


0.31  342 
0.31  310 
0.31  278 
0.31  246 
0.31  214 
0.31  382 


9.95  397 
9.95  391 
9.95  384 
9.95  378 
9.95  372 
9.95  366 


log  tan  I  log  sin 


6 
6 
6 
6 
6 

6 
6 
6 
6 
6 

5 
6 
6 
6 
6 

6 
6 
6 
6 
6 

6 
6 
6 
6 
6 

6 
6 
6 
6 
6 

6 
6 
6 
6 
6 

6 
6 
7 
6 
6 

6 
6 
6 
6 
6 

6 
6 
6 
6 

7 

6 
6 
6 
6 
6 

6 
7 
6 
6 
6 

dT 


33  32 


«3.3 
3^9 

4.4 
S.o 

s.s 

II.O 

30ji6.5 

40|22.0 

50:27.5 


3-2 

3-7 
4.3 
4.8 
5.3 
10.7 
16.0 
21.3 
26.7 


"  27  26 


2.7 
3.2 
3.6 
4.1 

4-5 
9.0 
3o;i3.S 
40 1 18.0 

50I22.S 


2.6 

3.0 

3.S 
3.9 

4-3 
8.7 
13.0 
17-3 
21.7 


6|  6 

0.6,0.5 
o.7;o.6 
0.8  |o.  7 
o.g  0.8 
1.00.8 
2.o'i.7 
30,3.513.0  2.5 
40  4.7I4.0  3.3 
50.5.815.0:4.2 


6  0.7 
7I0.8 
8J0.9 
9  I.I 
Io;i.2 
20  2.3 


Prop.  Pts. 


54 


Logarithms  of  the  Trigonometric  Functions 


26° 

# 

116°         206^         *296''        1 

loir  sill 

d. 

log  tan 

c.  d. 

log  cot 

log  cos 

d. 

Prop.  Pts. 

9.64   184 

26 

9.68  818 

0.31   182 

9.95  366 

6 
5 

60 

1 

9.64  210 

9.68  850 

32 
32 

0.31   150 

9.95  360 

59 

2 

9.64  236 

26 

9.68  882 

0.31   118 

9.95  354 

6 

58 

3 

9.64  262 

26 

9.68  914 

0.31   086 

9.95  348 

57 

4 

9.64-288 

25 

26 
26 
26 

9.68  9+6 

32 

0.31   054 

9.95   341 

6 

6 
6 

56 

55 

5 

9.64  313 

9.68  978 

0.31  022 

9.95  335 

6 

9.64  339 

9.69  010 

0.30  990 

9.95  329 

54 

7 

9.64  365 

9.69  0+2 

32 

0.30  958 

9.95  323 

6 

53 

S 

9.64  391 

20 
26 

9.69  074 

0.30  926 

9.95  317 

52 

9 

9.64  417 

25 

26 
26 

9.69   106 

32 

0.30  894 

9.95  310 

6 

6 
5 

51 
50 

ft 

10 

9.64  442 

9.69   138 

0.30  862 

9.95  304 

11 

9.64  468 

9.69   170 

32 

0.30  830 

9.95  298 

49 

6 

3-2 

3-1 

12 

9.64  494 

9.69   202 

32 

0.30   798 

9.95   292 

5 

48 

7 
8 
g 

3.7 

3.6 

13 

9.64  519 

26 

9.69   234 

0.30  766 

9.95  286 

47 

,1S 

4.6 

14 

9.64  545 

26 

9.69   266 

32 

0.30   734 

9.95   279 

6 

46 

10 

5-3 

5-2 

15 

9.64  571 

9.69   298 

0.30   702 

9.95   273 

6 
6 

7 
6 

45 

30 

16 

9.64  596 

25 

26 

9.69  329 

31 

0.30  671 

9.95   267 

44 

40 

1.320.7                 1 

17 

9.64  622 

9.69  361 

0.30  639 

9.95   261 

43 

50 

6.7,25.8                 1 

IS 

9.64  647 

25 

9.69   393 

32 
32 

0.30  607 

9.95   254 

42 

19 
20 

9.64  673 

25 

9.69  425 
9.69  457 

0.30  575 

9.95   248 

6 
6 

41 
40 

9.64  698 

0.30  543 

9.95  242 

21 

9.64  724 

26 

9.69  488 

31 

0.30  512 

9.95   236 

39 

22 

9.64  749 

25 

26 

9.69  520 

32 

0.30  480 

9.95   229 

7 

38 

23 

9.64  775 

9.69  552 

32 
31 

0.30  448 

9.95  223 

5 

37 

24 
25 

9.64  800 

26 

9.69  584 

0.30  416 

9.95  217 

6 

36 
35 

9.64  826 

9.69  615 

0.30  385 

9.95   211 

26 

9.64  851 

25 

26 

9.69  647 

32 

0.30  353 

9.95  204 

7 
6 
6 
7 
6 

6 

34 

27 

9.64  877 

9.69  679 

32 

0.30  321 

9.95   198 

33 

28 

9.64  902 

25 

26 

9  69  710 

32 
32 

0.30  290 

9.95   192 

32 

6 

2. 

5    2.S 

2.4 

29 
30 

9.64  927 

9.69   742 

0.30  258 

9.95   185 

31 
30 

7 
8 
9 

3. 
3.. 

3.< 

3     2.9 

s  3.3 

3     3.8 

2.8 
3.2 
3.6 

9.64  953 

9.69  774 

0.30  226 

9.95   179 

31 

9.64  978 

'= 

9.69  805 

31 

0.30  195 

9.95   173 

29 

10 

4- 

5    4-2 

4.0 

32 

9.65  003 

25 
26 

9.69  837 

32 

0.30  163 

9.95   167 

28 

20 

30 
40 

8. 
I3.< 
17. 

7    8.3 

8.0 

■33 

9.65  029 

9.69  868 

32 
32 

0.30  132 

9.95   160 

7 
5 

27 

J  16.7  16.0              1 

34 

35 

9.65  054 

25 

9.69  900 

0..30  100 

9.95   154 

6 

26 
25 

SO 

21. 

71 20.8 1 20.0              1 

9.65  079 

9.69  932 

0.30  068 

9.95   148 

36 

9.65   104 

25 

26 

9.69  963 

31 

0.30  037 

9.95   141 

7 
6 
6 

24 

37 

9.65   130 

9.69  995 

32 

0.30  005 

9.95   135 

23 

38 

9.65   155 

9.70  026 

0.29  974 

9.95   129 

22 

39 
40 

9.65   180 

25 

9.70  058 

*i 

0.29  942 

9.95   122 

6 

21 
20 

9.65   205 

9.70  089 

0.29  911 

9.95   116 

41 

9.65   230 

25 

9.70   121 

32 

0.29  879 

9.95   110 

6 

19 

42 

9.65  255 

25 

26 

9.70   152 

31 

0.29  848 

9.95   103 

7 
6 

18 

43 

9.65  281 

9-70   184 

0.29  816 

9.95  097 

17 

44 
45 

9.65  306 

25 

9.70  215 

32 

0.29  785 

9.95  090 

6 

16 
15 

6 

4 

0-7 

0 

0.6 

9.65  331 

9.70  247 

0.29  753 

9.95  084 

46 

9.65  356 

25 

9.70  278 

31 

0.29  722 

9.95  078 

14 

7I 

0.8 

0.7 
0  8 

47 

9.65  381 

25 

9.70  309 

31 

0.29  691 

9.95  071 

7 
6 
6 

7 

13 

9 

T.O 

0.9 

48 

9.65  406 

9.70  341 

0.29  659 

9.95  065 

12 

10 

1.2 

I.O 

49 
50 

9.65  431 

23 

25 

9.70  372 
9.70  404 

31 
32 

0.29  628 

9.95  059 

11 
10 

20 
30 
40 

2.3 

3-5 
4-7 

2.0 
3.0 
4.0 

9.65  456 

0.29  596 

9.95  052 

51 

9.65  481 

25 

9.70  435 

31 

0.29  565 

9.95  0+6 

6 

9 

1° 

5-8 

S-o 

52 

9.65   506 

25 

9.70  466 

31 

0.29  534 

9.95  039 

7 
6 
6 
7 

8 

53 

9.65   531 

25 

9.70  498 

32 

0.29  502 

9.95  033 

7 

54 

55 

9.65  556 

25 

24 

9.70  529 

31 
31 

0.29  471 

9.95  027 
9.95  020 

6 

5 

9.65   580 

9.70  560 

0.29  440 

56 

9.65  605 

25 

9.70  592 

32 

0.29  408 

9.95  014 

6 

4 

57 

9.65  630 

25 

9.70  623 

31 

0.29  377 

9.95  007 

7 
6 
6 

7 

3 

58 

9.65  655 

25 

9.70  654 

31 

0.29  346 

9.95  001 

2 

59 
60 

9.65  680 

25 
25 

9.70  685 

31 
32 

0.29  315 

9.94  995 

1 
0 

9.65   705 

9.70  717 

0.29  283 

9.94  988 

log  cos 

d. 

log  cot 

c.  d. 

log  tail 

log  sill 

d. 

/ 

Prop.  Pts. 

*I53° 

243° 

*333° 

63° 

1 

55 


Logarithms  of  the  Trigonometric  Fuxctiosts 


27° 

*ii 

7'        207°        *297°         1 

"o" 

log  sin 

d. 

log  tan 

c.  d.    log  cot  1 

log  cos 

d. 

Prop.  Pts. 

9.65   705 

9.70  717 

0.29  283 

9.94  988 

6 

60 

1 

9.65   729 

25 
25 

9.70  748 

0.29  252 

9.94  982 

59 

2 

9.65   754 

9.70  779 

31 
31 
32 

0.29  221 

9.94  975 

6 

58 

3 

9.65   779 

9.70  810 

0.29  190 

9.94  969 

57 

4 
5 

9.65  804 

24 

9.70  841 

0.29  159 

9.94  962 

6 

56 
55 

9.65  828 

9.70  873 

0.29  127 

9.94  956 

6 

9.65  853 

25 

25 
24 

25 

25 

9.70  904 

31 

0.29  096 

9.94  949 

7 
5 

54 

7 

9.65  878 

9.70  935 

31 
31 
31 

0.29  065 

9.94  943 

7 
6 

53 

8 

9.65  902 

9.70  966 

0.29  034 

9.94  936 

52 

9 
10 

9.65  927 

9.70  997 

0.29  003 

9.94  930 

7 

6 

6 

7 
6 

51 
50 

n 

82    31    30           1 

9.65  952 

9.71  028 

0.28  972 

9.94  923 

11 

9.65  976 

9.71  059 

31 

0.28  941 

9.94  917 

49 

6 

7 
8 

3-2 

3-7    . 
4-3 

5.1     3.0 
5-6    3-5 
i.i    4.0, 

12 

9.66  001 

24 

25 

25 

9.71  090 

0.28  910 

9.94  911 

48 

13 

9.66  025 

9.71  121 

32 
31 

0.28  879 

9.94  904 

47 

9 

4.8 

^.6    4-5 

14 

9.66  050 

9.71  153 

0.28  847 

9.94  898 

7 
6 

46 

45 

10 
20 
30 

5-3    5-21   5-0 
10.7  10.3' lO.O 

16.0  15.5 15.0 

15 

9.66  075 

9.71  184 

0.28  816 

9.94  891 

16 

9.66  099 

9.71  215 

31 

0.28  785 

9.94  885 

44 

40 

21.3120.7120.0 

17 

9.66  124 

9.71  246 

0.28  754 

9.94  878 

7 
6 

43 

....... 

18 

9.66  148 

25 

24 

9.71  277 

31 
31 

0.28  723 

9.94  871 

42 

19 
20 

9.66  173 

9.71  308 

0.28  692 

9.94  865 

7 
6 

41 
40 

9.66  197 

9.71  339 

0.28  661 

9.94  858 

21 

9.66  221 

24 

9.71  370 

31 

0.28  630 

9.94  852 

39 

22 

9.66  246 

25 

9.71  401 

31 

0.28  599 

9.94  845 

7 
6 
7 
6 

38 

23 

9.66  270 

25 

24 

9.71  431 

31 
31 

0.28  569 

9.94  839 

37 

24 
25 

9.66  295 

9.71  462 

0.28  538 

9.94  832 

36 
35 

9.66  319 

9.71  493 

0.28  507 

9.94  826 

26 

9.66  343 

24 

9.71  524 

31 

0.28  476 

9.94  819 

7 
6 

34 

ff 

25 

24    28 

27 

9.66  368 

9.71  555 

0.28  445 

9.94  813 

33 

28 

9.66  392 

9.71   586 

0.28  414 

9.94  806 

32 

6 

2.5 

2.4   2.3 

29 

9.66  416 

25 

9.71  617 

31 

0.28  383 

9.94  799 

6 

31 
30 

7 
8 
9 

2.9 
3.3 
3.8 

3-2    3-1 
3.6    3.S 

30 

9.66  441 

9.71  648 

0.28  352 

9.94  793 

31 
32 

9.66  465 
9.66  489 

24 

24 

9.71  679 
9.71  709 

31 
30 

0.28  321 
0.28  291 

9.94  786 
9.94  780 

7 
6 

29 

28 

10 
20 
30 

4.2  4.0    3-8            1 

8.3  8.0    7-7            1 

12.5  12.0  II. 5             1 

33 

9.66  513 

9.71  740 

0.28  260 

9.94  773 

27 

40 

16.7  I 

6.o|iS.3            1 

34 

9.66  537 

25 

9.71   771 

31 

0.28  229 

9.94  767 

7 

26 
25 

50|20.8|20.0{I9.2 

35 

9.66  562 

9.71  802 

0.28  198 

9.94  760 

36 

9.66  586 

24 

9.71  833 

31 

0.28  167 

9.94  753 

7 
6 

24 

■ 

37 

9.66  610 

24 

9.71  863 

30 

0.28  137 

9.94  747 

23 

38 

9.66  634 

24 

9.71  894 

31 

0.28  106 

9.94  740 

7 

22 

39 
40 

9.66  658 

24 

9.71  925 

30 

0.28  075 

9.94  734 

7 

21 
20 

9.66  682 

9.71  955 

0.28  045 

9.94  727 

41 

9.66  706 

24 

9.71  986 

31 

0.28  014 

9.94  720 

7 
6 

19 

42 

9.66  731 

25 

9.72  017 

31 

0.27  983 

9.94  714 

18 

43 

9.66  755 

24 

9.72  048 

0.27  952 

9.94  707 

17 

44 

9.66  779 

24 
24 

9.72  078 

30 

31 

0.27  922 

9.94  700 

7 
6 

16 

15 

"  7 

60. 

6 

!  0.6 

45 

9.66  803 

9.72  109 

0.27  891 

9.94  694 

46 

9.66  827 

24 

9.72  140 

31 

0.27  860 

9.94  687 

7 

14 

7o.i 
8o.( 
9  I. 

50.7 
)o.8 
[  0.9 

47 

9.66  851 

24 

9.72  170 

30 

0.27  830 

9.94  680 

7 
6 

13 

48 

9.66  875 

24 

9.72  201 

0.27  799 

9.94  674 

12 

10  I. 

2  I.O 

49 

9.66  899 

24 

9.72  231 

30 
31 

0.27   769 

9.94  667 

7 

11 

20  2. 

52.0 

50 

9.66  922 

9.72  262 

0.27  738 

9.94  660 

10 

404. 

74.0 

51 

9.66  946 

24 

9.72  293 

31 

0.27  707 

9.94  654 

9 

SOS- 

5S.o 

52 

9.66  970 

24 

9.72  323 

30 

0.27  677 

9.94  647 

7 

8 

53 

9.66  994 

24 

9.72  354 

31 

0.27  646 

9.94  640 

7 
6 
7 

7 

54 

9.67  018 

24 
24 

9.72  384 

30 

31 

0.27  616 

9.94  634 

6 

5 

55 

9.67  042 

9.72  415 

0.27  585 

9.94  627 

56 

9.67  066 

24 

9.72  445 

30 

0.27  555 

9.94  620 

7 
6 

4 

57 

9.67  090 

24 

9.72  476 

31 

0.27  524 

9.94  614 

3 

58 

9.67   113 

23 

9.72  506 

30 

0.27  494 

9.94  607 

2 

59 
60 

9.67   137 

24 
24 

9.72  537 

31 
30 

0.27  463 

9.94  600 

7 
7 

1 
0 

9.67   161 

9.72  567 

0.27  433 

9.94  593 

log  cos 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

f 

Prop.  Pts. 

*I52° 

242° 

*332° 

62° 

1 

66 


Logarithms  of  tiik  Trigonometric  Functions 


28° 

*ii8=       208^       *298"        1 

/ 

log  siu 

<1. 

log  tan 

c.  d. 

log  cot 

log  COS 

d. 

Prop.  Pts. 

0 

9.67   161 

9.72  567 

0.27  433 

9.94  593 

6 

60 

1 

9.67   185 

9.72  598 

0.27  402 

9.94  587 

59 

2 

9.67  208  1    ;t 

9.72  628 

31 

0.27  372 

9.94  580 

58 

3 

9.67  232 

9.72  659 

0.27  341 

9.94  573 

57 

4 

9.67  256 

24 

9.72  689 

31 

0.27  311 

9.94  567 

56 

55 

5 

9.67  280 

9.72  720 

0.27  280 

9.94  560 

6 

9.67  303 

23 

9.72  750 

30 

0.27  250 

9.94  553 

54 

7 

9.67  327 

23 

9.72  780 

31 

0.27  220 

9.94  546 

53 

8 

9.67  350 

9.72  811 

0.27   189 

9.94  540 

52 

9 

9.67  374 

24 

9.72  841 

31 

0.27   159 

9.94  533 

51 
50 

// 

31  1  so 

29 

10 

9.67  398 

9.72  872 

0.27   128 

9.94  526 

11 

9.67  421 

9.72  902 

0.27  098 

9.94  519 

49 

6 

3.t 

3.6 

4.1 

3.0 

2.9 

12 

9.67  445 

9.72  932 

30 
31 
30 
30 

0.27  068 

9.94  513 

48 

7 
8 

3.b 
4.0 

3.4 
3-9 

13 

9.67  468 

9.72  963 

0.27  037 

9.94  506 

47 

9 

4.6 

4.."; 

4.4 

14 

9.67  492 

23 

9.72  993 

0.27  007 

9.94  499 

46 

45 

10 

20 
30 

5.2 
10.3 

5.0 

lO.O 

i.i.o 

4.8 
9.7 
14.S 

15 

9.67  515 

9.73  023 

0.26  977 

9.94  492 

16 

9.67  539 

9.73  054 

0.26  946 

9.94  485 

44 

40 

20.7 

20.0 

19.3 

17 

9.67  562 

9.73  084 

0.26  916 

9.94  479 

43 

5025.01 25.0 

I24.2 

IS 

9.67  586 

23 

24 

9.73   114 

30 
31 

0.26  886 

9.94  472 

42 

19 

9.67  609 

9.73   144 

0.26  856 

9.94  465 

41 
40 

20 

9.67  633 

9.73   175 

0.26  825 

9.94  458 

21 

9.67  656 

23 

9.73   205 

30 

0.26  795 

9.94  451 

39 

22 

9.67  680 

24 

9.73   235 

30 

0.26  765 

9.94  445 

38 

23 

9.67   703 

23 
24 

9.73   265 

0.26  735 

9.94  438 

37 

24 

25 

9.67  726 

9.73  295 

31 

0.26  705 

9.94  431 

36 
35 

9.67  750 

9.73  326 

0.26  674 

9.94  424 

26 

9.67  773 

23 

9.73  356 

30 

0.26  644 

9.94  417 

34 

// 

04      Oft 

22 

27 

9.67  796 

23 

9.73  386 

30 

0.26  61'1 

9.94  410 

33 

28 

9.67  820 

9.73  416 

0.26  584 

9.94  404 

32 

6 

2.4 

2.3 

2.2 

29 

9.67  843 

23 

9.73  446 

30 

0.26  554 

9.94  397 

31 
30 

7 
8 
9 

2.8 
3.2 
3.6 

2.7 
3.1 

3.S 

2.6 
2.9 
3.3 

30 

9.67  866 

9.73  476 

0.26  524 

9.94  390 

31 

9.67  890 

24 

9.73  507 

31 

0.26  493 

9.94  383 

29 

10 

4.0 
8.0 
12.0 

3.8 

3.7 

32 

9.67  913 

23 

9.73  537 

30 

0.26  463 

9.94  376 

28 

30 

IT   S 

1 1.0 

33 

9.67  936 

9.73  567 

0.26  433 

9.94  369 

27 

40 

16.0 

I. '5.3 

14.7 

34 
35 

9.67  959 

23 

9.73  597 

30 

0.26  403 

9.94  362 

26 

25 

SO 

20.0 

19.2 

18.3 

9.67  982 

9.73  627 

0.26  373 

9.94  355 

36 

9.68  006 

24 

9.73  657 

30 

0.26  343 

9.94  349 

24 

37 

9.68  029 

23 

9.73  687 

30 

0.26  313 

9.94  342 

23 

38 

9.68  052 

9.73  717 

0.26  283 

9.94  335 

22 

39 

9.68  075 

23 

9.73  747 

30 

0.26  253 

9.94  328 

21 
20 

40 

9.68  098 

9.73  777 

0.26  223 

9.94  321 

41 

9.68   121 

23 

9.73  807 

30 

0.26  193 

9.94  314 

19 

42 

9.68  144 

23 

9.73  837 

30 

0.26  163 

9.94  307 

18 

43 

9.68   167 

9.73  867 

0.26  133 

9.94  300 

17 

44 
45 

9.68  190 

23 

9.73  897 

30 

0.26   103 

9.94  293 

16 

15 

6 

4 

0.7  c 

0 

.6 

9.68  213 

9.73  927 

0.26  073 

9.94  286 

46 

9.68  237 

24 

9.73  957 

30 

0.26  043 

9.94  279 

14 

7 

0.8  c 

.7 
g 

47 

9.68  260 

23 

9.73  987 

30 

0.26  013 

9.94  273 

13 

9 

T.O  C 

.9 

48 

9.68  283 

23 

9.74  017 

30 

0.25  983 

9.94  266 

12 

10 

1.2  I 

.0 

49 

9.68  305 

23 

9.74  047 

30 
30 

0.25  953 

9.94  259 

11 
10 

20 
30 
40 

2.32 
3.53 

.0 
.0 
.0 

50 

9.68  328 

9.74  077 

0.25  923 

9.94  252 

51 

9.68  351 

23 

9.74   107 

30 

0.25   893 

9.94  245 

9 

SO 

s.ttlj 

.0 

52 

9.68  374 

23 

9.74  137 

30 

0.25   863 

9.94  238 

8 

53 

9.68  397 

23 

9.74  166 

29 

0.25   834 

9.94  231 

7 

54 

9.68  420 

23 
23 

9.74  1% 

30 
30 

0.25  804 

9.94  224 

6 

5 

3^) 

9.68  443 

9.74  226 

0.25   774 

9.94  217 

56 

57 

9.68  466 
9.68  489 

23 

23 

9.74  256 
9.74  286 

30 
30 

0  25   744 

9.94  210 
9.94  203 

4 
3 

0.25   714 

58 

9.68  512 

23 

9.74  316 

30 

0.25  684 

9.94  196 

2 

59 
60 

9.68  534 

23 

9.74  345 

29 
30 

0.25  655 

9.94  189 

1 
0 

9.68  557 

9.74  375 

0.25   625 

9.94   182 

log  COS 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*i5i° 

241° 

*33i° 

61° 

1 

57 


Logarithms  of  the  Trigonometric  Functions 


29° 

* 

[19°    209^    *299°    1 

/ 

log  sin 

d. 

log  tan 

c.  d. 

log  cot 

log  cos 

d. 

Prop.  Pfs. 

0 

9.68  557 

9.74  375 

0.25  625 

9.94  182 

60 

1 

9.68  580 

9.74  405 

30 

0.25  595 

9.94  175 

59 

2 

9.68  603 

9.74  43i 

30 

0.25  565 

9.94  168 

7 

58 

3 

9.68  625 

9.74  465 

0.25  535 

9.94  161 

57 

4 

9.68  648 

23 

9.74  494 

30 

0.25  506 

9.94  154 

7 

56 

55 

5 

9.68  671 

9.74  524 

0.25  476 

9.94  147 

6 

9.68  694 

9.74  554 

0.25  446 

9.94  140 

7 

7 

54 

7 

9.68  716 

23 

9.74  583 

30 

0.25  417 

9.94  133 

53 

8 

9.68  739 

9.74  613 

0.25  387 

9.94  126 

52 

9 

9.68  762 

22 

9.74  643 

30 

0.25  357 

9.94  119 

7 

51 

"  30  1  »>« 

10 

9  68  784 

9.74  673 

0.25  327 

9.94  112 

60 

11 

9.68  807 

23 

9.74  702 

29 

0.25  298 

9.94  105 

49 

6  3 

0  2.9     1 

12 

9.68  829 

9.74  732 

0.25  268 

9.94  098 

8 

48 

73.SI  3.5 

8  4.0  3.9 

9  4.5  4.4 

13 

9.68  852 

23 
22 

9.74  762 

29 
30 

0.25  238 

9.94  090 

47 

14 

9.68  875 

9.74  791 

0.25  209 

9.94  083 

7 

46 

10  5.0  4.8 

15 

9.68  897 

9.74  821 

0.25  179 

9.94  076 

7 

7 
7 

45 

30  15 

014.5     1 

16 

9.68  920 

23 

9.74  851 

30 

0.25  149 

9.94  069 

44 

40  20.0  19.3      1 

17 

9.68  942 

9.74  880 

30 
29 
30 

0.25  120 

9.94  062 

43 

5025 

.0124.2    1 

18 

9.68  965 

9.74  910 

0.25  090 

9.94  055 

42 

19 
20 

9.68  987 

23 

9.74  939 

0.25  061 

9.94  048 

7 

41 
40 

9.69  010 

9.74  969 

0.25  031 

9.94  041 

21 

9.69  032 

9-74  998 

0.25  002 

9.94  034 

39 

22 

9.69  055 

9.75  028 

0.24  972 

9.94  027 

7 
8 

38 

23 

9.69  077 

9.75  058 

29 
30 

0.24  942 

9.94  020 

37 

24 

9.69  100 

22 

9.75  087 

0.24  913 

9.94  012 

7 

36 
35 

2.S 

9.69  122 

9.75  117 

0.24  883 

9.94  005 

26 

27 

9.69  144 
9.69  167 

23 

9.75  146 
9.75  176 

29 
30 

0.24  854 
0.24  824 

9.93  998 
9.93  991 

7 
7 
7 
7 

34 
33 

I"  2J 

t  22 

28 

9.69  189 

9.75  205 

0.24  795 

9.93  984 

32 

6  2 

3  2.2 

29 

9.69  212 

22 

9.75  235 

29 

0.24  765 

9.93  977 

31 

7  2 

8  3 

7  2.6 

30 

9.69  234 

9.75  264 

0.24  736 

9.93  970 

30 

9  3 

4  3.3 

31 

9.69  256 

22 

9-75  294 

30 

0.24  706 

9.93  963 

7 
8 

7 
7 

7 

29 

10  3 

8.3.7 

32 

9.69  279 

23 

9.75  323 

0.24  677 

9.93  955 

28 

20  7 

7  7-3 

33 

9.69  301 

9.75  353 

0.24  647 

9.93  948 

27 

40  IS 

3  14.7 

34 

9.69  323 

22 

9.75  382 

29 

0.24  618 

9.93  941 

26 

25 

50  19 

2  18.3 

35 

9.69  345 

9.75  411 

0.24  589 

9.93  934 

36 

9.69  368 

23 

9.75  441 

30 

0.24  559 

9.93  927 

7 

24 

37 

9.69  390 

9.75  470 

29 

0.24  530 

9.93  920 

7 
8 

7 
7 

23 

38 

9.69  412 

9.75  500 

30 

0.24  500 

9.93  912 

22 

39 
40 

9.69  434 

22 

9.75  529 

29 
29 

0.24  471 

9.93  905 

21 
20 

9.69  456 

9.75  558 

0.24  442 

9.93  898 

41 

9.69  479 

23 

9.75  588 

30 

0.24  412 

9.93  891 

7 

19 

42 

9.69  501 

22 

9.75  617 

29 

0.24  383 

9.93  884 

7 
8 

18 

43 

9.69  523 

9.75  647 

30 

0.24  353 

9.93  876 

17 

44 

9.69  545 

22 

9.75  676 

29 
29 

0.24  324 

9.93  869 

7 

16 

15 

6c 

>.8  0.7 

45 

9.69  567 

9.75  705 

0.24  295 

9.93  862 

46 

9.69  589 

22 

9.75  735 

30 

0.24  265 

9.93  855 

7 
8 

14 

7« 

8 

9 

>.9  0.8 

47 

9.69  611 

22 

9.75  764 

29 

0.24  236 

9.93  847 

13 

.2  I.O 

48 

9.69  633 

9.75  793 

29 

0.24  207 

9.93  840 

12 

10 

.3  1-2 

49 

9.69  655 

9.75  822 

29 

0.24  178 

9.93  833 

7 

11 

20 

i.^  2.3 

50 

9.69  677 

9.75  852 

0.24  148 

9.93  826 

10 

40 

;.3  4.7 

51 

9.69  699 

22 

9.75  881 

29 

0.24  119 

9.93  819 

7 

9 

50  ( 

5.7  s-S 

52 

9.69  721 

22 

9.75  910 

29 

0.24  090 

9.93  811 

8 

8 

53 

9.69  743 

9.75  939 

29 

0.24  061 

9.93  804 

7 

7 

54 

9.69  765 

22 

9.75  969 

30 
29 

0.24  031 

9.93  797 

8 

6 

5 

55 

9.69  787 

9.75  998 

0.24  002 

9.93  789 

56 

9.69  809 

22 

9.76  027 

29 

0.23  973 

9.93  782 

7  - 

4 

57 

9.69  831 

22 

9.76  056 

29 

0.23  944 

9.93  775 

7 

3 

58 

9.69  853 

22 

9.76  086 

30 

0.23  914 

9.93  768 

7 
8 
7 

2 

59 
60 

9.69  875 

22 

9.76  115 

29 
29 

0.23  885 

9.93  760 

1 
0 

9.69  897 

9.76  144 

0.23  856 

9.93  753 

log  COS 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

L_ 

Prop.  Pts. 

*I50°     2 

J40° 

*33o° 

60° 

1 

58 


Logarithms  op  the  Trigoxometric  Functions 


30° 

*I20 

^   210^   *30o^ 

/ 

log  sin 

d. 

log  tau 

c.  d. 

log  cot 

log  COS 

d. 

Prop.  Pts. 

0 

y.69  897 

9.76  144 

'  0.23  856 

9.93  753 

60 

1 

9.69  919 

22 

9.76  173 

0.23  827 

9.93  746 

59 

2 

9.69  941 

9.76  202 

29 

0.23  798 

9.93  738 

58 

3 

9.69  963 

9.76  231 

0.23  769 

9.93  731 

57 

4 

9.69  984 

22 

9.76  261 

29 

0.23  739 

9.93  724 

7 

8 

56 

55 

5 

9.70  006 

9.76  290 

0.23  710 

9.93  717 

6 

9.70  028 

9.76  319 

0.23  681 

9.93  709 

54 

7 

9.70  050 

9.76  348 

29 

0.23  652 

9.93  702 

53 

S 

9.70  072 

9.76  377 

0.23  623 

9.93  695 

8 

52 

9 

9.70  093 

22 

9.76  406 

29 

0.23  594 

9.93  687 

' 

51 
50 

"  SO 

id    28    1 

10 

9.70  115 

9.76  435 

0.23  565 

9.93  680 

11 

9.70  137 

9.76  464 

29 

0.23  536 

9.93  673 

7 

8 

7 
8 

49 

6  3.01  2.9,  2.8 

7  3.s|  3.4|  3-3 

8  4-OI  3-9  3.7 

12 

9.70  159 

9.76  493 

29 
29 
29 

0.23  507 

9.93  665 

48 

13 

9.70  180 

9.76  522 

0.23  478 

9.93  658 

47 

9  4.5  4-4]  4-2 

14 

9.70  202 

22 

9.76  551 

0.23  449 

9.93  650 

7 

46 
45 

10  5.o|  4.8|  4-7 

20  lO.O;  9.7  9.3 

30  15.0  I4.s'l4.0 

15 

9.70  224 

9.76  580 

0.23  420 

9.93  643 

16 

9.70  245 

9.76  609 

29 

0.23  391 

9.93  636 

8 

44 

40  20.0  19.3I18.7 

17 

9.70  267 

9.76  639 

0.23  361 

9.93  628 

43 

.,-..-.-... 

IS 

9.70  288 

9.76  668 

29 
28 

0.23  332 

9.93  621 

7 
8 

42 

19 

9.70  310 

22 

9.76  697 

0.23  303 

9.93  614 

41 
40 

20 

9.70  332 

9.76  725 

0.23  275 

9.93  606 

21 

9.70  353 

9.76  754 

0.23  246 

9.93  599 

8 

39 

22 

9.70  375 

9.76  783 

0.23  217 

9.93  591 

38 

23 

9.70  3% 

9.76  812 

29 
29 

0.23  188 

9.93  584 

7 
8 

37 

24 

9.70  418 

21 

9.76  841 

0.23  159 

9.93  577 

36 
35 

25 

9.70  439 

9.76  870 

0.23  130 

9.93  569 

26 

9.70  461 

9.76  899 

0.23  101 

9.93  562 

8 

34 

"  < 

a   21 

27 

9.70  482 

9.76  928 

0.23  072 

9.93  554 

33 

28 

9.70  504 

9.76  957 

0.23  043 

9.93  547 

g 

32 

6 

2.2  2.1 

29 

9.70  525 

22 

9.76  986 

29 

0.23  014 

9.93  539 

7 

31 
30 

7 
8 
9 

Z.6  2.4 
2.9  2.8 
3-3  3-2 

30 

9.70  547 

9.77  015 

0.22  985 

9.93  532 

31 

9.70  568 

9.77  044 

29 

0.22  956 

9.93  525 

7 
8 

29 

10 

J.7  3-5 

32 

9.70  590 

9.77  073 

29 
28 

0.22  927 

9.93  517 

28 

30  I 

i.o  10.5 

33 

9.70  611 

9.77  101 

0.22  899 

9.93  510 

8 

7 

27 

40  I 

4.714.0 

34 

9.70  633 

21 

9.77  130 

29 

0.22  870 

9.93  502 

26 
25 

501 

8.3.17.S 

35 

9.70  654 

9.77  159 

0.22  841 

9.93  495 

36 

9.70  675 

9.77  188 

29 

0.22  812 

9.93  487 

8 

24 

37 

9.70  697 

9.77  217 

29 

0.22  783 

9.93  480 

7 
8 

23 

38 

9.70  718 

9.77  246 

28 
29 

0.22  754 

9.93  472 

22 

39 

9.70  739 

22 

9.77  274 

0.22  726 

9.93  465 

8 

21 
20 

40 

9.70  761 

9.77  303 

0.22  697 

9.93  457 

41 

9.70  782 

9.77  332 

29 

0.22  668 

9.93  450 

7 

19 

42 

9.70  803 

9.77  361 

29 

0.22  639 

9.93  442 

18 

43 
44 

9.70  824 
9.70  846 

22 
21 

9.77  390 
9.77  418 

29 
28 
29 

0.22  610 
0.22  582 

9.93  435 
9.93  427 

7 
8 

7 

17 
16 
15 

6< 

8  7 
3.8  0.7 

45 

9.70  867 

9.77  447 

0.22  553 

9.93  420 

46 

9.70  888 

21 

9.77  476 

29 

0.22  524 

9.93  412 

8 

14 

7< 

3.9  0.8 

47 

9.70  909 

9.77  505 

29 
28 

0.22  495 

9.93  405 

7 
8 

13 

9 

.2  1.0 

48 

9.70  931 

9.77  533 

0.22  467 

9.93  397 

12 

10 

.3  1-2 

49 

9.70  952 

21 

9.77  562 

29 

0.22  438 

9.93  390 

8 

11 
10 

20 

30. 

40 

2-712.3 

J-ol3-S 

;-3;4-7 

50 

9.70  973 

9.77  591 

0.22  409 

9.93  382 

51 

9.70  994 

21 

9.77  619 

28 

0.22  381 

9.93  375 

7 

9 

50I 

).7l5-8 

52 

9.71  015 

9.77  648 

29 

0.22  352 

9.93  367 

8 

8 

53 

9.71  036 

9.77  677 

29 

0.22  323 

9.93  360 

7 
8 
8 

7 

54 

9.71  058 

21 

9.77  706 

29 
28 

0.22  294 

9.93  352 

6 

5 

55 

9.71  079 

9.77  734 

0.22  266 

9.93  344 

56 

9.71  100 

21 

9.77  763 

29 

0.22  237 

9.93  337 

7 

4 

3/ 

9.71  121 

21 

9.77  791 

0.22  209 

9.93  329 

8 

3 

58 

9.71  142 

9.77  820 

29 

0.22  180 

9.93  322 

7 

2 

59 

9.71  163 

21 

9.77  849 

29 

28 

0.22  151 

9.93  314 

8 
7 

1 
0 

60 

9.71  184 

9.77  877 

0.22  123 

9.93  307 

log  cos 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*I49= 

239° 

*329" 

59° 

59 


Logarithms  of  the  Trigonomktbic  Functions 


31° 

*I2l''     211°     *30I°      1 

1 

log  sin 

d. 

log  tan 

c.  d. 

log  cot 

log  cos 

d. 

Prop.  Pts. 

0 

y.71  184 

9.77  877 

0.22  123 

9.93  307 

8 

8 

7 
8 

60 

1 

9.71  205 

^ 

9.77  906 

0.22  094 

9.93  299 

59 

2 

9.71  226 

9.77  935 

28 

0.22  065 

9.93  291 

58 

3 

9.71  247 

9.77  963 

29 
28 

0.22  037 

9.93  284 

57 

4 
5 

9.71  268 

21 

9.77  992 

0.22  008 

9.93  276 

7 

8 
8 

56 

55 

9.71  289 

9.78  020 

0.21  980 

9.93  269 

6 

9.71  310 

9.78  049 

28 
29 
29 
28 

0.21  951 

9.93  261 

54 

7 

9.71  331 

9.78  077 

0.21  923 

9.93  253 

53 

8 

9.71  352 

9.78  106 

0.21  894 

9.93  246 

8 

52 

9 
10 

9.71  373 

20 

9.78  135 

0.21  865 

9.93  238 

8 

51 
50 

t/ 

29  ^<*      1 

9.71  393 

9.78  163 

0.21  837 

9.93  230 

11 

9.71  414 

9.78  192 

29 

28 

0.21  808 

9.93  223 

7 
8 
8 
7 
8 

49 

6 

2.9 

2.8 

12 

9.71  435 

9.78  220 

0.21  780 

9.93  215 

48 

7 
8 
9 

3.i 

3-3 

13 

9.71  456 

9.78  249 

28 

0.21  751 

9.93  207 

47 

4.4 

4.2 

14 

9.71  477 

21 

9.78  277 

29 

0.21  723 

9.93  200 

46 

10 

4.8 

4.7 

15 

9.71  498 

9.78  306 

28 

0.21  694 

9.93  192 

8 

45 

30 

14.  S 

14.0 

16 

9.71  519 

9.78  334 

0.21  666 

9.93  184 

44 

40 

19-3 

18.7 

17 

9.71  539 

9.78  363 

29 
28 

0  21  637 

9.93  177 

7 
8 
8 

7 

43 

SO 

24.2 

23-3 

18 

9.71  560 

9.78  391 

28 

0.21  609 

9.93  169 

42 

19 

9.71  581 

21 

9.78  419 

29 

28 

0.21  581 

9.93  161 

41 
40 

20 

9.71  602 

9.78  448 

0.21  552 

9.93  154 

21 

9.71  622 

9.78  476 

0.21  524 

9.93  146 

8 

39 

22 

9.71  643 

9.78  505 

28 

0.21  495 

9.93  138 

38 

23 

9.71  664 

21 

9.78  533 

0.21  467 

9.93  131 

7 
8 
8 

37 

24 
25 

9.71  685 

20 

9.78  562 

28 

0.21  438 

9.93  123 

36 
35 

9.71  705 

9.78  590 

0.21  410 

9.93  115 

26 

9.71  726 

9.78  618 

28 

0.21  382 

9.93  108 

7 

34 

27 

9.71  747 

9.78  647 

29 
28 

0.21  353 

9.93  100 

8 
8 
7 
8 
8 
8 

33 

" 

'Zl 

zu 

28 

9.71  767 

9.78  675 

0.21  325 

9.93  092 

32 

6 

2.1 

2.0 

29 
30 

9.71  788 

21 

9.78  704 

28 

0.21  296 

9.93  084 

31 
30 

7 
8 
9 

2.4 
2.8 
S.7 

2.3 
2.7 

3.0 

9.71  809 

9.78  732 

0.21  268 

9.93  077 

31 

9.71  829 

9.78  760 

0.21  240 

9.93  069 

29 

10 

3-5 

3.3 

32 

9.71  850 

9.78  789 

29 
28 
28 

0.21  211 

9.93  061 

28 

20 

7.0 

6.7 

33 

9.71  870 

9.78  817 

0.21  183 

9.93  053 

7 
8 

8 
8 
8 

27 

40 

T/|.0 

13-3 

34 
35 

9.71  891 

20 

9.78  845 

29 

0.21  155 

9.93  046 

26 

25 

sol 

I7.S 

16.7 

9.71  911 

9.78  874 

0.21  126 

9.93  038 

36 

9.71  932 

9.78  902 

28 

0.21  098 

9.93  030 

24 

37 

9.71  952 

9.78  930 

0.21  070 

9.93  022 

23 

38 

9.71  973 

9.78  959 

28 
28 

0.21  041 

9.93  014 

7 
8 

8 
8 

22 

39 

9.71  994 

20 

9.78  987 

0.21  013 

9.93  007 

21 
20 

40 

9.72  014 

9.79  015 

0.20  985 

9.92  999 

41 

9.72  034 

20 

9.79  043 

0.20  957 

9.92  991 

19 

42 

9.72  055 

9.79  072 

29 
28 
28 
28 

0.20  928 

9.92  983 

18 

43 

9.72  075 

9.79  100 

0.20  900 

9.92  976 

8 

17 

44 
45 

9.72  096 

20 

9.79  128 

0.20  872 

9.92  968 

8 

8 
8 
8 

16 
15 

( 

)o.8 

7 

0.7 

9.72  116 

9.79  156 

0.20  844 

9.92  960 

46 

9.72  137 

21 

9.79  185 

29 

0.20  815 

9.92  952 

14 

i 

0.9 

0.8 

47 

9.72  157 

9.79  213 

28 
28 

0.20  787 

9.92  944 

13 

48 

9.72  177 

9.79  241 

0.20  759 

9.92  936 

12 

IC 

)  1.3 

1.2 

49 

9.72  198 

9.79  269 

0.20  731 

9.92  929 

8 

11 

2C 

)2.7 

2.3 

3c 
4<: 

)  4.0 
>  5-3 

50 

9.72  218 

9.79  297 

0.20  703 

9.92  921 

10 

4-7 

51 

9.72  238 

20 

9.79  326 

29 

0.20  674 

9.92  913 

8 
8 
8 
8 
8 

9 

S« 

>b.7 

S.8 

52 

9.72  259 

21 

9.79  354 

0.20  646 

9.92  905 

8 

53 

9.72  279 

9.79  382 

28 
28 

0.20  618 

9.92  897 

7 

54 

55 

9.72  299 

21 

9.79  410 

0.20  590 

9.92  889 

6 

5 

9.72  320 

9.79  438 

0.20  562 

9.92  881 

56 

9.72  340 

20 

9.79  466 

28 

0.20  534 

9.92  874 

7 

4 

57 

9.72  360 

20 

9.79  495 

29 

0.20  505 

9.92  866 

8 
8 
8 

3 

58 

9.72  381 

9.79  523 

28 
28 

0.20  477 

9.92  858 

2 

59 
60 

9.72  401 

20 

9.79  551 

0.20  449 

9.92  850 

1 
0 

9.72  421 

9.79  579 

0.20  421 

9.92  842 

log  cos 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*I48° 

238° 

*328^ 

58° 

1 

60 


Logarithms  of  the  Trigonometric  Functions 


32° 

* 

122=     212=     *302°     1 

1 

log-  sin 

(1. 

log  tan 

c.  d. 

log  cot 

log  cos 

d. 

Prop.  Pts. 

0 

9.72  421 

9.79  579 

28 

0.20  421 

9.92  842 

8 

60 

1 

9.72  441 

9.79  607 

28 

0.20  393 

9.92  834 

8 

59 

2 

9.72  461 

9.79  635 

28 

0.20  365 

9.92  826 

8 

58 

3 

9.72  482 

20 

9.79  663 

28 

0.20  337 

9.92  818 

8 

57 

4 

5 

9.72  502 

20 

9.79  691 

28 

0.20  309 

9.92  810 

7 

8 

8 

56 

55 

9.72  522 

9.79  719 

0.20  281 

9.92  803 

6 

9.72  542 

20 

9.79  747 

29 
28 

0.20  253 

9.92  795 

54 

/ 

9.72  562 

9.79  776 

0.20  224 

9.92  787 

8 

55 

8 

9.72  582 

20 

9.79  804 

28 

0.20  196 

9.92  779 

8 

52 

9 
10 

9.72  602 
9.72  622 

20 

9.79  832 

28 

28 
28 
28 

0.20  168 

9.92  771 

8 

8 
8 
8 

51 
50 

"  2 

9.79  860 

0.20  140 

9.92  763 

11 

9.72  643 

9.79  888 

0.20  112 

9.92  755 

49 

6  2 

.9  2.8 

2.7 

12 

9.72  663 

9.79  916 

0.20  084 

9.92  747 

48 

7  3 

8  3 

9  4 

.5  3.3 

3.2 
|3.6 

13 

9.72  683 

9.79  944 

28 

0.20  056 

9.92  739 

8 

47 

.4  4.2 

14 
15 
16 

9.72  703 

20 
20 

9.79  972 

28 

28 
28 
28 

0.20  028 

9.92  731 

8 

8 
8 
8 

46 
45 
44 

10  4.8  4.7 
20  9.7  9.3 
30  14.S  14.0 
40  19.3  18.7 

4.5 
1  9-0 
13.S 
18.0 

9.72  723 
9.72  743 

9.80  000 
9.80  028 

0.20  000 
0.19  972 

9.92  723 
9.92  715 

17 

9.72  763 

9.80  056 

0.19  944 

9.92  707 

43 

5024 

.2I23.3 

22.5 

18 

9.72  783 

9.80  084 

28 

0.19  916 

9.92  699 

8 

42 

19 
20 

9.72  803 

20 

9.80  112 
9.80  140 

28 
28 

0.19  888 

9.92  691 

8 

8 
8 
8 

41 
40 

9.72  623 

0.19  860 

9.92  683 

21 

9.72  843 

9.80  168 

0.19  832 

9.92  675 

39 

22 

9.72  863 

9.80  195 

27 
28 
28 

0.19  805 

9.92  667 

38 

23 

9.72  883 

19 
20 

9.80  223 

0.19  777 

9.92  659 

8 

37 

24 

9.72  902 

9.80  251 

28 

28 
28 
28 
28 

0.19  749 

9.92  651 

8 

8 
8 
8 

36 
35 

25 

9.72  922 

9.80  279 

0.19  721 

9.92  643 

26 

9.72  942 

9.80  307 

0.19  693 

9.92  635 

34 

"  21 

20 

19 

27 

9.72  962 

9.80  335 

0.19  665 

9.92  627 

33 

28 

9.72  982 

9.80  363 

0.19  637 

9.92  619 

8 

32 

6  2. 

I  2.0 

1-9 

29 

9.73  002 

-20 

9.80  391 

28 
28 

0.19  609 

9.92  611 

8 

8 
8 
8 
8 

31 
30 

7  2. 

8  2. 

9  3- 

4  2.3 

8  2.7 
2  3.0 

2.2 

2.S 
2.9 

30 

9.73  022 

9.80  419 

0.19  581 

9.92  603 

31 

9.73  041 

9.80  447 

0.19  553 

9.92  595 

29 

10  3- 

s  3.3 

3.2 

32 

9.73  061 

9.80  474 

27 
28 
28 

0.19  526 

9.92  587 

28 

20  7. 

0  6.7 

6.3 

33 

9.73  081 

9.80  502 

0.19  498 

9.92  579 

27 

40  14. 

0  13.3 

12.7 

34 

9.73  101 

20 

9.80  530 

28 

0.19  470 

9.92  571 

8 

26 

25 

sol  17. 

5  16.7 

15.8 

35 

9.73  121 

9.80  558 

0.19  442 

9.92  563 

36 

9.73  140 

19 

9.80  586 

28 
28 
28 

0.19  414 

9.92  555 

8 

24 

37 

9.73  160 

9.80  614 

0.19  386 

9.92  546 

9 

8 
8 
8 

23 

38 

9.73  180 

9.80  642 

0.19  358 

9.92  538 

22 

39 

9.73  200 

19 

9.80  669 

28 

0.19  331 

9.92  530 

21 
20 

40 

9.73  219 

9.80  697 

0.19  303 

9.92  522 

41 

9.73  239 

9.80  725 

28 
28 
28 

0.19  275 

9.92  514 

8 

19 

42 

9.73  259 

9.80  753 

0.19  247 

9.92  506 

8 
8 
8 

18 

43 

9.73  278 

9.80  781 

0.19  219 

9.92  498 

17 

1 

44 
45 

9.73  298 

20 

9.80  808 

28 

0.19  192 

9.92  490 

16 
15 

6c 

9  8 

.90.8 

4 

0.7 

9.73  318 

9.80  836 

0.19  164 

9.92  482 

46 

9.73  337 

19 

9.80  864 

28 

0.19  136 

9.92  473 

9 

14 

7  I 

8  I 

.110.9 

0.9 
I.I 

47 

9.73  357 

9.80  892 

28 

0.19  108 

9.92  465 

8 
8 
8 
8 

13 

9  I 

.4  1.2 

48 

9.73  377 

9.80  919 

27 
28 
28 

0.19  081 

9.92  457 

12 

10  1 

■  5  1-3 

1.2 

49 

9.73  396 

20 

9.80  947 

0.19  053 

9.92  449 

11 
10 

20  3 
4o|( 

.0  2.7 

[-5  4.0 
.0  5-3 

2-3 

3-5 
4-7 

50 

9.73  416 

9.80  975 

0.19  025 

9.92  441 

51 

9.73  435 

19 

9.81  003 

28 

0.18  997 

9.92  433 

8 

9 

sol' 

.50.7 

S.8 

52 

9.73  455 

20 

9.81  030 

27 

0.18  970 

9.92  425 

8 

8 

53 

9.73  474 

19 

9.81  058 

28 
27 

0.18  942 

9.92  416 

9 

8 
8 

7 

54 

9.73  494 

19 

9.81  086 

0.18  914 

9.92  408 

6 

5 

55 

9.73  513 

9.81  113 

0.18  887 

9.92  400 

56 

9.73  533 

20 

9.81  141 

28 

0.18  859 

9.92  392 

8 

4 

.■)/ 

9.73  552 

19 

9.81  169 

28 

0.18  831 

9.92  384 

8 

3 

58 

9.73  572 

20 

9.81  196 

27 

0.18  804 

9.92  376 

2 

59 

9.73  591 

19 

20 

9.81  224 

28 
28 

0.18  776 

9.92  367 

9 

8 

1 
0 

60 

9.73  611 

9.81  252 

0.18  748 

9.92  359 

log  COS 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*I47° 

237° 

*327° 

57° 

1 

fil 


Logarithms  of  the  Trigonometric  Functions 


33° 

*i 

23' 

213°        *303° 

log  sin 

d. 

log  tan  ! 

c.  d.    log  cot  1 

log  cos 

d. 

Prop.  Pts. 

9.73  611 

9.81   252 

0.18   748 

9.92  359 

8 

60 

1 

9.73  630 

9.81   279 

28 

0.18   721 

9.92  351 

59 

2 

9.73  650 

19 

9.81   307 

28 

0.18  693 

9.92  343 

g 

58 

3 

9.73  669 

9.81   33i 

0.18  665 

9.92  335 

9 

8 

8 
8 

57 

4 

9.73  689 

19 

9.81   362 

28 
28 

0.18  638 

9.92  326 

56 

55 

5 

9.73  708 

9.81   390 

0.18  610 

9.92  318 

6 

9.73  727 

19 

9.81   418 

0.18   582 

9.92  310 

54 

7 

9.73   747 

20 

9.81   445 

28 

0.18   555 

9.92  302 

53 

8 

9.73   766 

19 
20 

9.81   473 

0.18   527 

9.92  293 

8 

52 

9 

9.73  785 

9.81   500 

18 
28 

0.18  500 

9.92  285 

8 
8 

51 
50 

// 

28 

2; 

so 

10 

9.73  805 

9.81   528 

0.18  472 

9.92  277 

11 

9.73  824 

19 

9.81   556 

0.18  -144 

9.92  269 

49 

6 
7 
8 

2. 

t    i- 

7    2.0 
2    2.3 
6    2.7 

12 

9.73  843 

9.81   583 

28 

0.18  417 

9.92  260 

g 

48 

3-7l  3. 

13 

9.73  863 

19 
19 

9.81   611 

27 
28 

0.18  389 

9.92  252 

g 

47 

9 

4. 

2I   4 

I    3.0 

14 

9.73  882 

9.81   638 

0.18  362 

9.92  244 

9 

8 
8 

g 

46 
45 

10 
20 
30 

4-71   4-5    3-3 
9.3    9.0    6.7 
14.0  13. s  lo.o 

15 

9.73  901 

9.81   666 

0.18  334 

9.92  235 

16 

9.73  921 

19 

9.81   693 

27 

0.18  307 

9.92  227 

44 

40 

18. 

7  IS 

013.3             1 

.La  ..             1 

17 

9.73  940 

19 

9.81    721 

0.18  279 

9.92  219 

43 

18 

9.73  959 

19 

9.81    748 

28 

0.18  252 

9.92  211 

9 
8 

8 

42 

19 

9.73  978 

19 

9.81    776 

27 
28 

0.18   224 

9.92  202 

41 
40 

20 

9.73  997 

9.81   803 

0.18   197 

9.92  194 

21 

9.74  017 

9.81   831 

0.18  169 

9.92  186 

39 

22 

9.74  036 

19 

9.81   858 

28 

0.18  142 

9.92  177 

38 

23 

9.74  055 

19 

9.81   886 

27 
28 

0.18  114 

9.92  169 

g 

37 

24 

25 

9.74  074 

19 

9.81   913 

0.18  087 

9.92  161 

9 

8 
8 

36 
35 

9.74  093 

9.81   941 

0.18  059 

9.92   152 

26 

9.74  113 

9.81   968 

27 
28 

0.18  032 

9.92  144 

34 

// 

19 

18 

27 

9.74  132 

9.81   996 

0.18  004 

9.92  136 

33 

28 

9.74  151 

19 

9.82  023 

28 

0.17  977 

9.92  127 

g 

32 

6 

1.9 

1.8 

29 

9.74  170 

19 

9.82  051 

27 
28 

0.17  949 

9.92   119 

8 

31 
30 

7 
8 
9 

2.5 
2.9 

2.4 
2.7 

30 

9.74   189 

9.82  078 

0.17  922 

9.92   111 

31 

9.74  208 

9.82   106 

0.17  894 

9.92  102 

9 

8 

29 

10 

3-2 

6.3 

9.,'i 

3.0 
6  0 

32 

9.74  227 

9.82   133 

28 

0.17  867 

9.92  094 

28 

30 

9.0 

33 

9.74  246 

19 
19 

9.82   161 

0.17  839 

9.92  086 

27 

40 

2.7 

12.0 

34 

9.74  265 

9.82   188 

27 
28 

0.17  812 

9.92  077 

8 

26 
25 

SO 

S-ii 

iS.o 

35 

9.74  284 

9.82   215 

0.17  785 

9.92  069 

36 

9.74  303 

19 

9.82   243 

0.17  757 

9.92  060 

9 

8 
8 

24 

37 

9.74  322 

9.82  270 

27 
28 

0.17  730 

9.92  052 

23 

38 

9.74  341 

19 
19 

9.82  298 

0.17  702 

9.92  044 

22 

39 
40 

9.74  360 

9.82  325 

27 

0.17  675 

9.92  035 

8 

21 
20 

9.74  379 

9.82   352 

0.17  648 

9.92  027 

41 

9.74  398 

19 

9.82  380 

28 

0.17  620 

9.92  018 

9 

8 
8 
9 
8 

19 

42 

9.74  417 

19 

9.82  407 

27 
28 
27 
27 

0.17  593 

9.92  010 

18 

43 
44 

9.74  436 
9.74  455 

19 
19 

9.82  435 
9.82   462 

0.17  565 
0.17  538 

9.92  002 
9.91  993 

17 
16 

15 

6 

9 

0.9 

8 

D.8 

45 

9.74  474 

9.82  489 

0.17   511 

9.91  985 

46 

9.74  493 

19 

9.82  517 

0.17  483 

9.91  976 

9 

8 

14 

7 
g 

I.I 

3.9 

47 

9.74  512 

19 

9.82  544 

27 

0.17  456 

9.91  968 

13 

9 

1.4 

1.2 

48 

9.74  531 

18 
19 

9.82  571 

27 
28 
27 

0.17  429 

9.91  959 

8 
9 

12 

10 

1-5 

t.3 

49 

9.74  549 

9.82  599 

0.17  401 

9.91  951 

11 
10 

20 
30 
40 

3.0 
4-5 
6.0 

2.7 
4.0 

5-3 

50 

9.74  568 

9.82   626 

0.17  374 

9.91  942 

51 

9.74  587 

19 

9.82  653 

27 

0.17  347 

9.91  934 

8 

9 

SO 

7-5 

6.7 

52 

9.74  606 

19 

9.82  681 

28 

0.17  319 

9.91  925 

9 
8 

8 

53 

9.74  625 

19 

9.82   708 

27 

0.17  292 

9.91  917 

7 

54 

9.74  644 

18 

9.82   735 

27 
27 

0.17  265 

9.91  908 

8 

6 

5 

55 

9.74  662 

9.82  762 

0.17  238 

9.91  900 

56 

9.74  681 

19 

9.82  790 

28 

0.17  210 

9.91  891 

9 

8 

4 

57 

9.74  700 

19 

9.82  817 

27 

0.17   183 

9.91  883 

3 

58 

9.74  719 

19 

9.82  844 

27 

0.17  156 

9.91  874 

2 

59 

9.74  737 

19 

9.82  871 

28 

0.17   129 

9.91  866 

9 

1 
0 

60 

9.74  756 

9.82  899 

0.17  101 

9.91  857 

log  COS 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*I46° 

236°        *326° 

56° 

Logarithms  of  the  Trigonometric  Functions 


34° 

*I24°   214°   *304'^    1 

/ 

log  sin 

(1. 

log  tan 

c.d. 

log  cot 

log  cos 

d. 

Prop.  Pts. 

0 

9.74  756 

9.82  899 

0.17  101 

9.91  857 

8 

60 

1 

9.7+  775 

9.82  926 

0.17  074 

9.91  849 

59 

2 

9.74  794 

i8 

9.82  953 

27 
28 

0.17  047 

9.91  840 

8 

58 

3 

9.74  812 

9.82  980 

0.17  020 

9.91  832 

57 

4 

9.74  831 

19 
i8 

9.83  008 

27 

0.16  992 

9.91  823 

8 

56 

55 

5 

9.74  850 

9.83  035 

0.16  965 

9.91  815 

6 

9.74  868 

9.83  062 

0.16  938 

9.91  806 

a 

54 

7 

9.74  887 

i8 

9.83  089 

28 

0.16  911 

9.91  798 

53 

8 

9.74  906 

9.83  117 

0.16  883 

9.91  789 

8 

52 

9 

9.74  924 

19 
i8 

9.83  144 

27 

0.16  856 

9.91  781 

9 

51 
50 

" 

28 

27  j  26 

10 

9.74  943 

9.83  171 

0.16  829 

9.91  772 

11 

9.74  961 

9.83  198 

27 

0.16  802 

9.91  763 

8 

9 

8 

49 

6 

2.8 

2.7;  2.6 

12 

9.74  980 

19 
19 

18 

9.83  225 

27 

0.16  775 

9.91  755 

48 

^ 

3.J 

3.2  3.0 
3.6  3.5 

13 

9.74  999 

9.83  252 

28 

0.16  748 

9.91  746 

47 

9 

4.2 

4-0  3-9 

14 

9.75  017 

19 

9.83  280 

27 

0.16  720 

9.91  738 

9 

46 

10 

4-7 

4-5  4.3 

15 

9.75  036 

18 

9.83  307 

0.16  693 

9.91  729 

45 

30 

14.C 

13-5  I3-0 

16 

9.75  054 

9.83  334 

0.16  666 

9.91  720 

8 

44 

40 

18.7 

18.0  17.3 

17 

9.75  073 

18 

9.83  361 

0.16  639 

9.91  712 

43 

SO  23.3 

22.5,21.7 

18 

9.75  091 

19 
18 

9.83  388 

0.16  612 

9.91  703 

8 

42 

19 

9.75  110 

9.83  415 

27 

0.16  585 

9.91  695 

9 

41 
40 

20 

9.75  128 

9.83  442 

0.16  558 

9.91  686 

21 

9.75  147 

18 

9.83  470 

0.16  5.30 

9.91  677 

8 

39 

22 

9.75  165 

9.83  497 

0.16  503 

9.91  669 

38 

23 

9.75  184 

18 

9.83  524 

0.16  476 

9.91  660 

37 

24 
25 

9.75  202 

19 
18 

9.83  551 

27 

0.16  449 

9.91  651 

8 

36 
35 

9.75  221 

9.83  578 

0.16  422 

9.91  643 

26 

9.75  239 

9.83  605 

27 

0.16  395 

9.91  634 

34 

„ 

19  18 

27 

9.75  258 

9.83  632 

0.16  368 

9.91  625 

33 

28 

9.75  276 

18 

9.83  659 

27 

0.16  341 

9.91  617 

32 

6 

1.9  1.8 

29 

9.75  294 

19 

18 

9.83  686 

27 

0.16  314 

9.91  608 

9 

31 
30 

7 
8 
9 

2.2  2.1 

2.5j  2.4 
2.8t  2.7 

30 

9.75  313 

9.83  713 

0.16  287 

9.91  599 

31 

9.75  331 

9.83  740 

27 

0.16  260 

9.91  591 

29 

10 

3-2:  3.0 

32 

9.75  350 

19 

18 

9.83  768 

0.16  232 

9.91  582 

9 

28 

30 

9.5;  9.0 

33 

9.75  368 

18 

9.83  795 

0.16  205 

9.91  573 

8 

27 

40  I 

2.7  I2.0 

34 
35 

9.75  386 

19 

18 
18 
18 

9.83  822 

27 

0.16  178 

9.91  565 

9 

26 

25 

SOI 

S-8:iS-o 

9.75  405 

9.83  849 

0.16  151 

9.91  556 

36 

9.75  423 

9.83  876 

27 

0.16  124 

9.91  547 

9 

24 

37 

9.75  441 

9.83  903 

r  '' 

0.16  097 

9.91  538 

9 
8 

23 

38 

9.75  459 

9.83  930 

0.16  070 

9.91  530 

22 

39 

9.75  478 

18 

9.83  957 

27 

0.16  043 

9.91  521 

9 

21 
20 

40 

9.75  496 

9.83  984 

0.16  016 

9.91  512 

41 

9.75  514 

9.84  Oil 

27 

0.15  989 

9.91  504 

19 

42 

9.75  533 

19 
18 
18 
18 

9.84  038 

27 

0.15  962 

9.91  495 

9 

18 

43 

44 
45 

9.75  551 
9.75  569 
9.75  587 

9.84  065 
9.84  092 

27 
27 

0.15  935 
0.15  908 

9.91  486 
9.91  477 

9 
8 

17 
16 
15 

6 

9 

0.9 

8 

0.8 

9.84  119 

0.15  881 

9.91  469 

46 

9.75  605 

18 

9.84  146 

27 

0.15  8.54 

9.91  460 

9 

14 

7 

1.0 

0.9 

47 

9.75  624 

19 
18 
18 
18 

9.84  173 

27 

0.15  827 

9.91  451 

9 

13 

g 

1.4 

1.2 

48 

9.75  642 

9.84  200 

0.15  800 

9.91  442 

12 

IC 

1. 5 

1.3 

49 

9.75  660 

9.84  227 

27 

0.15  773 

9.91  433 

8 

11 
10 

2C 
40 

3.0 
4-5 
6.0 

2.7 
4.0 

5-3 

50 

9.75  678 

9.84  254 

0.15  746 

9.91  425 

51 

9.75  696 

18 

9.84  280 

26 

0.15  720 

9.91  416 

9 

9 

SO 

7-5  0.7      1 

52 

9.75  714 

18 

9.84  307 

27 

0.15  693 

9.91  407 

9 

8 

53 

9.75  733 

19 
18 
iS 

9.84  334 

27 

0.15  666 

9.91  398 

9 

7 

54 

55 

9.75  751 

9.84  361 
9.84  388 

27 
27 

0.15  639 

9.91  389 

9 
8 

6 

5 

9.75  769 

0.15  612 

9.91  381 

56 

9.75  787 

18 

9.84  415 

27 

0.15  585 

9.91  372 

9 

4 

57 

9.75  805 

9.84  442 

27 

0.15  558 

9.91  363 

9 

3 

58 

9.75  823 

9.84  469 

27 

0.15  531 

9.91  354 

9 

2 

59 

9.75  841 

18 

9.84  496 

27 
27 

0.15  504 

9.91  345 

9 
9 

1 
0 

9.75  859 

9.84  523 

0.15  477 

9.91  336 

lo^  cos 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*i45° 

^35° 

*325° 

55° 

1 

63 


LOGABITHMS    OF   THE    TRIGONOMETRIC   FUNCTIONS 


35° 

*I25° 

215°   *305° 

f 

"o 

log  sin 

d. 

log  tan 

c.  d. 

log  cot 

log  cos 

d. 

Piop.  Pts. 

9.75  859 

9.84  523 

0.15  477 

9.91  336 

60 

1 

9.75  877 

i8 

9.84  550 

26 

0.15  450 

9.91  328 

9 

59 

2 

9.75  895 

i8 

9.84  576 

0.15  424 

9.91  319 

58 

3 

9.75  913 

i8 

9.84  603 

0.15  397 

9.91  310 

57 

4 

5 

9.75  931 

i8 

i8 
i8 
i8 

9.84  630 
9.84  657 

27 

0.15  370 

9.91  301 

9 

56 

55 

9.75  949 

0.15  343 

9.91  292 

6 

9.75  967 

9.84  684 

0.15  316 

9.91  283 

54 

7 

9.75  985 

9.84  711 

27 
26 

0.15  289 

9.91  274 

8 

53 

8 

9.76  003 

i8 

9.84  738 

0.15  262 

9.91  266 

52 

9 

9.76  021 

i8 

i8 
i8 
i8 
i8 

9.84  764 

27 

0.15  236 

9.91  257 

9 

51 
50 

// 

27 

26 

18 

10 

9.76  039 

9.84  791 

0.15  209 

9.91  248 

11 

9.76  057 

9.84  818 

0.15  182 

9.91  239 

49 

6 

2." 

2. 

5  1.8 

12 
13 

9.76  075 
9.76  093 

9.84  845 
9.84  872 

27 
27 
26 

0.15  155 
0.15  128 

9.91  230 
9.91  221 

9 

48 
47 

7 
8 
9 

3.2  3.0  2.1       1 
3.6  3.5  2.4       1 
4.1  3.0  2.7      1 

14 

9.76  111 

i8 

9.84  899 

0.15  101 

9.91  212 

9 

46 

10 

4.- 

4-3  3.0 

)  8.7  6.0 

13.0  9.0 

15 

9.76  129 

9.84  925 

0.15  075 

9.91  203 

45 

30 

13. 

16 

9.76  146 

i8 

9.84  952 

0.15  048 

9.91  194 

44 

40 

18.C 

)  17.3  12.0 

17 

9.76  164 

9-84  979 

0.15  021 

9.91  185 

43 

50 

22., 

izi. 

7I1S.0     1 

IS 

9.76  182 

i8 

9.85  006 

0.14  994 

9.91  176 

9 
9 

42 

19 

9.76  200 

i8 

9.85  033 

26 

0.14  967 

9.91  167 

41 
40 

20 

9.76  218 

9.85  059 

0.14  941 

9.91  158 

21 

9.76  236 

9.85  086 

27 

0.14  914 

9.91  149 

9 

39 

22 

9.76  253 

17 
i8 

9.85  113 

27 

0.14  887 

9.91  141 

38 

23 

9.76  271 

9.85  140 

26 

0.14  860 

9-91  132 

9 
9 

37 

24 

9.76  289 

i8 

9.85  166 
9.85  193 

27 

0.14  834 

9.91  123 

36 
35 

25 

9.76  307 

0.14  807 

9.91  114 

26 

9.76  324 

17 
i8 

9.85  220 

27 

0.14  780 

9.91  105 

9 

34 

27 

9.76  342 

9.85  247 

26 

0.14  753 

9.91  096 

33 

" 

IV 

10 

28 

9.76  360 

i8 

9.85  273 

0.14  727 

9.91  087 

9 
9 

32 

6 

T.7 

1.0 

29 
30 

9.76  378 

17 

i8 
i8 

9.85  300 

27 

0.14  700 

9.91  078 

31 
30 

7 
8 
g 

2.0 

2.3 
76 

1.2 
1.3 

9.76  395 

9.85  327 

0.14  673 

9.91  069 

31 

9.76  413 

9.85  354 

27 
26 

0.14  646 

9.91  060 

9 

29 

10 

2.8 

1.7 

32 

9.76  431 

9.85  380 

0.14  620 

9.91  051 

28 

20 

5-7 
8.5 

T.3 

3-3 

33 

9.76  448 

17 
18 

9.85  407 

27 

0.14  593 

9.91  042 

9 
9 
10 

27 

30 
40  ] 

S.o 
6.7 

34 
35 

9.76  466 

18 

9.85  434 

26 

0.14  566 

9.91  033 

26 

25 

SOI 

4.2 

8.3 

9.76  484 

9.85  460 

0.14  540 

9.91  023 

36 

9.76  501 

17 
18 
18 

9.85  487 

27 

0.14  513 

9.91  014 

9 

24 

37 

9.76  519 

9.85  514 

27 
26 

0.14  486 

9.91  005 

9 

23 

38 

9.76  537 

9.85  540 

0.14  460 

9.90  996 

22 

39 

9.76  554 

18 

9.85  567 

27 

0.14  433 

9.90  987 

9 

21 
20 

40 

9.76  572 

9.85  594 

0.14  406 

9.90  978 

41 

9.76  590 

18 

9.85  620 

26 

0.14  380 

9.90  969 

9 

19 

42 

9.76  607 

17 
i8 
17 
18 

9.85  647 

27 

0.14  353 

9.90  960 

9 

18 

43 
44 

45 

9.76  625 
9.76  642 

9.85  674 
9.85  700 

27 
26 

27 

0.14  326 
0.14  300 

9.90  951 
9.90  942 

9 
9 
9 

17 
16 

15 

6 

9 

O.Q 

8 

3.8 

9.76  660 

9.85  727 

0.14  273 

9.90  933 

46 

9.76  677 

17 

9.85  754 

27 

0.14  246 

9.90  924 

9 

14 

7 
8 

I.I 

3.9 

47 

9.76  695 

9.85  780 

26 

0.14  220 

9.90  915 

9 

13 

48 

9.76  712 

9.85  807 

27 

0.14  193 

9.90  906 

12 

10 

I..S 

1.3 

49 

9.76  730 

9.85  834 

0.14  166 

9.90  896 

11 

20 

3.0 

2.7 

50 

9.76  747 

9.85  860 

0.14  140 

9.90  887 

10 

40 

6.0 

5.3 

51 

9.76  765 

18 

9.85  887 

27 

0.14  113 

9.90  878 

9 

9 

50 

7.5 

6.7 

52 

9.76  782 

17 
18 

9.85  913 

26 

0.14  087 

9.90  869 

9 

8 

53 

9.76  800 

9.85  940 

27 

0.14  060 

9.90  860 

9 

7 

54 

9.76  817 

18 

9.85  967 

27 
26 

0.14  033 

9.90  851 

9 
9 

6 

5 

55 

9.76  835 

9.85  993 

0.14  007 

9.90  842 

56 

9.76  852 

17 

9.86  020 

27 

0.13  980 

9.90  832 

10 

4 

57 

9.76  870 

9.86  046 

26 

0.13  954 

9.90  823 

9 

3 

58 

9.76  887 

17 

9.86  073 

27 

0.13  927 

9.90  814 

9 

2 

59 
60 

9.76  904 

17 
18 

9.86  100 

27 
26 

0.13  900 

9.90  805 

9 

1 
0 

9.76  922 

9.86  126 

0.13  874 

9.90  796 

log  COS 

d. 

log  cot 

c.  d. 

log  tan   log  sin 

d. 

/ 

Prop.  Pts. 

*I44° 

234° 

*324° 

54° 

64 


Logarithms  of  the  Trigoxometric  Functioxs 


36° 

* 

26°    216^    *3o6°    1 

"o" 

log  sin 

d. 

log  tan  j 

c.  d. 

log  cot 

log  cos 

d. 

Prop.  rts. 

9.76  922 

9.86  126 

0.13  874 

9.90  796 

60 

1 

9.76  939 

i8 

9.86  153 

26 

0.13  847 

9.90  787 

59 

2 

9.76  957 

17 

9.86  179 

27 
26 

0.13  821 

9.90  777 

9 
9 
9 

58 

3 

9.76  974 

9.86  206 

0.13  794 

9.90  768 

57 

4 
5 

9.76  991 

i8 

9.86  232 
9.86  259 

27 
26 

0.13  768 

9.90  759 

56 
55 

9.77  009 

0.13  741 

9.90  750 

6 

9.77  026 

17 
i8 

9.86  285 

0.13  715 

9.90  741 

54 

7 

9.77  043 

9.86  312 

26 

0.13  688 

9.90  731 

:o 

53 

8 

9.77  061 

9.86  338 

27 
27 

26 

0.13  662 

9.90  722 

9 
9 

52 

9 
10 

9.77  078 

17 

9.86  365 

0.13  635 

9.90  713 

51 
50 

"  27 

26  18 

9.77  095 

9.86  392 

0.13  608 

9.90  704 

11 

9.77  112 

i8 

9.86  418 

0.13  582 

9.90  694 

49 

6  2.7 

2.6  1.8 

12 
13 

9.77  130 
9.77  147 

17 
17 

9.86  445 
9.86  471 

26 

0.13  555 
0.13  529 

9.90  685 
9.90  676 

9 
9 
9 
10 

48 
47 

7  3.2 

8  3.6 

9  4.0 

3.0  2.1 
3.5  2.4 
3.9  2.7 

14 

9.77  164 

17 
i8 

9.86  498 

26 

0.13  502 

9.90  667 

46 
45 

10  4.5 
20  9.0 
30  13.5 

4-3  3.0 
8.7  6.0 
13-0  9.0 

15 

9.77  181 

9.86  524 

0.13  476 

9.90  657 

16 

9.77  199 

17 
17 

9.86  551 

26 
26 

0.13  449 

9.90  648 

44 

40  18.0 

17.3  12.0 

17 

9-77  216 

9.86  577 

0.13  423 

9.90  639 

43 

50  22.5 

21.7  15.0 

18 

9.77  233 

9.86  603 

27 
26 

0.13  397 

9.90  630 

42 

19 
20 

9.77  250 

i8 

9. 86  630 

0.13  370 

9.90  620 

9 

41 
40 

9.77  268 

9.86  656 

0.13  344 

9.90  611 

21 

9.77  285 

9.86  683 

26 

0.13  317 

9.90  602 

9 

39 

22 

9.77  302 

17 

9.86  709 

0.13  291 

9.90  592 

38 

23 

9.77  319 

9.86  736 

26 

0.13  264 

9.90  583 

9 

37 

24 

25 

9.77  336 

17 

9.86  762 

27 

26 
27 

0.13  238 

9.90  574 

9 

36 

35 

9.77  353 

9.86  789 

0.13  211 

9.90  565 

26 
27 

9.77  370 

9.77  387 

17 
17 
i8 

9.86  815 
9.86  842 

0.13  185 
0.13  158 

9.90  555 
9.90  546 

10 
9 

34 
33 

"  1 

7  16 

28 

9.77  405 

17 
17 

9.86  868 

26 

0.13  132 

9.90  537 

9 

32 

6  I 

.7  1.6 

29 
30 

9.77  422 

9.86  894 

27 
26 

0.13  106 

9.90  527 

9 

31 
30 

7  2 

8  2 

9  2 

.0  1.9 

.3   2.1 

.6  2.4 

9.77  439 

9.86  921 

0.13  079 

9.90  518 

31 

9.77  456 

17 

9.86  947 

0.13  053 

9.90  509 

9 

29 

10  2 

.8  2.7 

32 
33 

9.77  473 
9.77  490 

17 
17 
17 

9.86  974 

9.87  000 

27 
26 
27 
26 

26 

0.13  026 
0.13  000 

9.90  499 
9.90  490 

9 

28 

27 

20  5 
30  8 
40  II 

•  7  5.3 
.5  8.0 
.3  I0.7 

34 

35 

9.77  507 

9.87  027 

0.12  973 

9.90  480 

9 

26 

25 

5014 

.2  13.3 

9.77  524 

9.87  053 

0.12  947 

9.90  471 

36 

9.77  541 

9.87  079 

0.12  921 

9.90  462 

9 

24 

37 

9.77  558 

9.87  106 

26 
26 

0.12  894 

9.90  452 

23 

38 

9.77  575 

17 
17 

9.87  132 

0.12  868 

9.90  443 

9 

22 

39 

9.77  592 

9.87  158 

27 
26 

0.12  842 

9.90  434 

10 

21 
20 

40 

9.77  609 

9.87  185 

0.12  815 

9.90  424 

41 

9.77  626 

17 

9.87  211 

0.12  789 

9.90  415 

9 

19 

42 

9.77  643 

17 

9.87  238 

27 
26 
26 
27 

0.12  762 

9.90  405 

10 

18 

43 

9.77  660 

9.87  264 

0.12  736 

9.90  396 

9 

17 

44 

9.77  677 

17 

9.87  290 

0.12  710 

9.90  386 

9 

16 

15 

"  1 

6  1 

0  V 

.0  0.9 

45 

9.77  694 

9.87  317 

0.12  683 

9.90  377 

46 

9.77  711 

17 

9.87  343 

26 

0.12  657 

9.90  368 

9 

14 

7  J 

8  1 

9  I 

.2  I.O 

47 

9.77  728 

17 
16 

9.87  369 

0.12  631 

9.90  358 

10 

13 

•  S  1-4 

48 

9.77  744 

9.87  396 

26 
26 

0.12  604 

9.90  349 

12 

10  ] 

.7  i-S 

49 
50 

9.77  761 

17 

9.87  422 

0.12  578 

9.90  339 

9 

11 
10 

20  i 
30  , 
40  ( 

•33.0 
.04.5 
).7  6.0 

9.77  778 

9.87  448 

0.12  552 

9.90  330 

51 

9.77  795 

17 

9.87  475 

27 

0.12  525 

9.90  320 

10 

9 

SO  i 

5.3  7.S 

52 

9.77  812 

17 

9.87  501 

26 

0.12  499 

9.90  311 

9 

8 

53 

9.77  829 

9.87  527 

0.12  473 

9.90  301 

7 

54 
55 

9.77  846 

16 

9.87  554 

.26 

0.12  446 
0.12  420 

9.90  292 

10 

6 

5 

9.77  862 

9.87  580 

9.90  282 

56 

9.77  879 

17 

9.87  606 

26 

0.12  394 

9.90  273 

9 

4 

57 

9.77  896 

17 

9.87  633 

27 
26 
26 
26 

0.12  367 

9.90  263 

3 

58 

9.77  913 

17 

9.87  659 

0.12  341 

9.90  254 

9 

2 

59 
60 

9.77  930 

17 
16 

9.87  685 

0.12  315 

9.90  244 

9 

1 
0 

9.77  946 

9.87  711 

0.12  289 

9.90  235 

log  COS 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

1 

Prop.  Pts. 

*I43° 

233^ 

*323° 

53° 

1 

65 


Logarithms  of  the  Tkigonometric  Functions 


37° 


^127' 


217" 


^307^ 


log  sin 


d. 


log  tan   c.  d.    log  cot 


log  cos 


d. 


Prop.  Pts. 


10 

11 
12 
13 
14 


20 

21 

22 
23 

^ 

25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


35 
36 
37 
38 

40 

41 
42 
43 
44 


9.77  946 
9.77  963 
9.77  980 

9.77  997 

9.78  013 


9.78  030 
9.78  047 
9.78  063 
9.78  080 
9.78  097 


9.78 
9.78 
9.78 
9.78 
9.78 


113 
130 
147 
163 

180 


9.78  197 
9.78  213 
9.78  230 
9.78  246 
9.78  263 


9.78  280 
9.78  296 
9.78  313 
9.78  329 
9.78  346 


9.78  362 
9.78  379 
9.78  395 
9.78  412 
9.78  428 


9.78  445 
9.78  461 
9.78  478 
9-78  494 
9.78  510 


9.78  527 
9.78  543 
9.78  560 
9.78  576 
9.78  592 


9.78  609 
9.78  625 
9.78  642 
9-78  658 
9.78  674 


9.78  691 
9.78  707 
9.78  723 
9.78  739 
9.78  756 


9.78  772 
9.78  788 
9.78  805 
9.78  821 
9.78  837 


9.78  853 
9.78  869 
9.78  886 
9.78  902 
9.78  918 
9.78  934 

log  COS 


^142'^ 


232^ 


9.87  711 
9.87  738 
9.87  764 
9.87  790 
9.87  817 


9.87  843 
9.87  869 
9.87  895 
9.87  922 
9.87  948 


9.87  974 

9.88  000 
9.88  027 
9.88  053 
9.88  079 


9.88  105 
9.88  131 
9.88  158 
9.88  184 
9.88  210 


9.88  236 
9.88  262 
9.88  289 
9.88  315 
9.88  341 


9.88  367 
9.88  393 
9.88  420 
9.88  446 
9.88  472 


9.88  498 
9.88  524 
9.88  550 
9.88  577 
9.88  603 


9.88  629 
9.88  655 
9.88  681 
9.88  707 
9.88  733 


9.88  759 
9.88  786 
9.88  812 
9.88  838 
9.88  864 


9.88  890 
9.88  916 
9.88  942 
9.88  968 
9.88  994 


9.89  020 
9.89  046 
9.89  073 
9.89  099 
9.89  125 


9.89  151 
9.89  177 
9.89  203 
9.89  229 
9.89  255 
9.89  281 


log  cot  c.  d. 


»322- 


27 
26 
26 
27 
26 

26 
26 
27 
26 
26 

26 
27 
26 
26 
26 

26 
27 
26 
26 
26 

26 
27 
26 
26 
26 

26 
27 
26 
26 
26 

26 
26 
27 
26 
26 

26 
26 
26 
26 
26 

27 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
27 
26 
26 
26 

26 
26 
26 
26 
26 


12  289 
12  262 
12  236 
12  210 
12  183 


9.90  235 
9.90  225 
9.90  216 
9.90  206 
9.90  197 


12  157 
12  131 
12  105 
12  078 
12  052 


9.90  187 

9.90  178 

9.90  168 

9.90  159 

9.90  149 


12  026 
12  000 
11  973 
11  947 
11  921 


9.90  139 
9.90  130 
9.90  120 
9.90  111 
9.90  101 


11  895 

11  869 

11  842 

11  816 

11  790 


9.90  091 
9.90  082 
9.90  072 
9.90  063 
9.90  053 


11  764 

11  738 

11  711 

11  685 

11  659 


9.90  043 
9.90  034 
9.90  024 
9.90  014 
9.90  005 


11  633 

11  607 

11  580 

11  554 

11  528 


9.89  995 
9.89  985 
9.89  976 
9.89  966 
9.89  956 


11  502 

11  476 

11  450 

11  423 

11  397 


9.89  947 
9.89  937 
9.89  927 
9.89  918 
9.89  908 


11  371 

11  345 

11  319 

11  293 

11  267 


9.89  898 
9.89  888 
9.89  879 
9.89  869 
9.89  859 


11  241 
11  214 
11  188 
11  162 
11  136 


9.89  849 
9.89  840 
9.89  830 
9.89  820 
9.89  810 


11  110 
11  084 
11  058 
11  032 
11  006 


9.89  801 
9.89  791 
9.89  781 
9.89  771 
9.89  761 


10  980 
10  954 
10  927 
10  901 
10  875 


9.89  752 
9.89  742 
9.89  732 
9.89  722 
9.89  712 


10  849 
10  823 
10  797 
10  771 
10  745 


■10  719 
log  tan 


9.89  702 
9.89  693 
9.89  683 
9.89  673 
9.89  663 
9.89  653 

log  sin 


52° 

66 


d. 


27 

26 

2.7 

2.6 

3.2 

.S-o 

.•5.6 

3.S 

4.0 

3.9 

4.S 

4..S 

9.0 

8.7 

i,?.5 

13.0 

18.0 

17.3 

22.S 

21.7 

1.7 

2.0 

2.3 

2.6 
2.8 

5.7 

8.5 
"3 
14.2 


16    10  I  9 


i.o  0.9 
1.2  i.o 
l.3|l.2 

I.S1I.4 
1.7  I. S 
3-3  3.0 
5-04.5 
6.7  6.0 
8.3I7.S 


Prop.  Pts. 


Logarithms  of  the  Trigonometric  Functions 


38° 

*I28 

'-■       aiS'^       *3o8'' 

/ 

log  siu 

d. 

log  tan 

c.  d. 

log  cot 

log  cos 

d. 

Prop.  Pts. 

0 

9.78  934 

i6 

9. 89  281 

26 

0.10  719 

9.89  653 

60 

1 

9.78  950 

9.89  307 

26 

0.10  693 

9.89  643 

10 

59 

2 

9.78  967 

i6 

9.89  333 

26 

0.10  667 

9.89  633 

9 

58 

3 

9.78  983 

i6 

9.89  359 

26 

0.10  641 

9.89  624 

57 

4 

9.78  999 

i6 
i6 

9.89  385 

26 

26 
26 

0.10  615 

9.89  614 

10 

56 

55 

5 

9.79  015 

9.89  411 

0.10  589 

9.89  604 

6 

9.79  031 

i6 

9.89  437 

0.10  563 

9.89  594 

10 

54 

" 

26 

25 

7 

9.79  047 

t6 

9.89  463 

26 

0.10  537 

9.89  584 

53 

6 

-'  6 

2.5 

8 

9.79  063 

t6 

9.89  489 

26 

0.10  511 

9.89  574 

52 

7 

3.0 

2.9 

9 

9.79  079 

i6 
i6 

9.89  515 

26 

26 
26 
26 

0.10  485 

9.89  564 

10 

51 
50 

8 
9 
10 

3-5 
3-9 
4-3 

i-i 
3.8 
4.2 

10 

9.79  095 

9.89  541 

0.10  459 

9.89  554 

11 

9.79   111 

17 
i6 

9.89  567 

0.10  433 

9.89  544 

10 

49 

20 

8.7 

8.3 

12 

9.79  128 

9.89  593 

0.10  407 

9.89  534 

48 

30 

13-" 

12-5 

16.7 

13 

9.79  144 

i6 

9.89  619 

26 

0.10  381 

9.89  524 

47 

50 

21.7 

20.8 

14. 

9.79  160 

i6 
i6 

9.89  645 

26 

26 
26 

0.10  355 

9.89  514 

10 

46 

15 

9.79  176 

9.89  671 

0.10  329 

9.89  504 

45 

16 

9.79  192 

i6 

9.89  697 

0.10  303 

9.89  495 

44 

17 

9.79  208 

i6 

9.89  723 

26 

0.10  277 

9.89  485 

10 

43 

18 

9.79  224 

i6 

9.89  749 

26 

0.10  251 

9.89  475 

42 

19 

9.79  240 

i6 
i6 

9.89  775 

26 

26 
26 
26 
26 

0.10  225 

9.89  465 

10 

41 
40 

"1  17   1  1« 

20 

9.79  256 

9.89  801 

0.10  199 

9.89  455 

21 

9.79  272 

i6 

9.89  827 

0.10  173 

9-89  445 

39 

6 

1-7 

1.6 

22 
23 

9.79  288 
9.79  304 

i6 
15 

9.89  853 
9.89  879 

0.10  147 
0.10  121 

9.89  435 
9.89  425 

10 

38 
37 

7 
8 
9 

2.0 
2.3 
2.6 

1.9 

2.1 
2.4 

24 

9.79  319 

16 

9.89  905 

26 

0.10  095 

9.89  415 

10 

36 

10 

2.8 

2.7 

25 

9.79  335 

16 

9.89  931 

26 
26 
36 

0.10  069 

9.89  405 

35 

30 

8.S 

8.0 

26 

9.79  351 

16 

9.89  957 

0.10  043 

9.89  395 

34 

40 

II.3 

10.7 

27 

9.79  367 

i6 

9.89  983 

0.10  017 

9.89  385 

33 

SO 

14.2 

13-3 

28 

9.79  383 

16 

9.90  009 

26 

0.09  991 

9.89  375 

32 

29 

9.79  399 

16 
16 

9.90  035 

26 

0.09  965 

9.89  364 

10 

31 
30 

30 

9.79  415 

9.90  061 

0.09  939 

9.89  354 

31 

9.79  431 

9.90  086 

2S 
26 
26 
26 

0.09  914 

9.89  344 

29 

32 

9.79  447 

16 

9.90  112 

0.09  888 

9.89  334 

28 

33 

9.79  463 

9.90  138 

0.09  862 

9  89  324 

27 

// 

15 

11 

34 
35 

9.79  478 

16 

16 
16 
16 

9.90  164 

26 

26 
26 
26 
26 

0.09  836 

9.89  314 

10 

26 

25 

6 

7 

l.S 

T  8 

I.I 

1.3 

9.79  494 

9.90  190 

0.09  810 

9.89  304 

36 

9.79  510 

9.90  216 

0.09  784 

9.89  294 

24 

8 

2.0 

i-i 

37 

9.79  526 

9.90  242 

0.09  758 

9.89  284 

23 

9 

2.2 

1.6 
1.8 
3.7 

38 

9.79  542 

16 

9.90  268 

0.09  732 

9.89  274 

22 

5.0 

39 

9.79  558 

15 

16 
i6 
16 

9.90  294 

26 
26 

0.09  706 

9.89  264 

10 
10 

21 

20 

19 

30 
40 
SO 

7-5 

lO.O 

12.5 

s-s 

7-3 
9.8 

40 

41 

9.79  573 
9.79  589 

9.90  320 
9.90  346 

0.09  680 
0.09  654 

9.89  254 
9.89  244 

42 

9.79  605 

9.90  371 

25 
26 
26 
26 

0.09  629 

9.89  233 

18 

43 

9.79  621 

9.90  397 

0.09  603 

9.89  223 

17 

44 

9.79  636 

16 

9.90  423 

0.09  577 

9.89  213 

10 

16 
15 

45 

9.79  652 

9.90  449 

0.09  551 

9.89  203 

46 

9.79  668 

16 

9.90  475 

0.09  525 

9.89  193 

14 

47 

9.79  684 

9.90  501 

26 
26 
26 
25 

0.09  499 

9.89  183 

13 

// 

10 

9 

48 

9.79  699 

15 
16 
16 

9.90  527 

0.09  473 

9.89  173 

12 

49 

9.79  715 

9.90  553 

0.09  447 

9.89  162 

10 

• 

11 
10 

7 
8 

1.2 
1-3 

0.9 

I.O 

1.2 

50 

9.79  731 

9.90  578 

0.09  422 

9.89  152 

51 

9.79  746 

IS 

9.90  604 

26 

0.09  396 

9.89  142 

10 

9 

9 

i-S 
1.7 
S.3 

1.4 
i.S 
3.0 

52 

9.79  762 

9.90  630 

0.09  370 

9. 89  132 

10 

8 

20 

53 

9.79  778 

9.90  656 

26 
26 

0.09  344 

9.89  122 

7 

30 

5.0 

4-5 

54 

9.79  793 

15 
16 

9.90  682 

0.09  318 

9.89  112 

II 

6 

5 

40 
50 

(J.  7 
8.^ 

7.5 

55 

9.79  809 

9.90  708 

0.09  292 

9.89  101- 

56 

9.79  825 

16 

9.90  734 

26 

0.09  266 

9. 89  091 

10 

4 

57 

9.79  840 

IS 

9.90  759 

25 

0.09  241 

9.89  081 

10 

3 

58 

9.79  856 

9.90  785 

26 
26 

0.09  215 

9.89  071 

2 

59 

9.79  872 

15 

9.90  811 

0.09  189 

9.89  060 

10 

1 
0 

60 

9.79  887 

9.90  837 

0.09  163 

9.89  050 

log  COS 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*i4i°         : 

^31° 

*32I° 

51°                                           1 

67 


LOGAKITHMS    OF    THE    TRIGONOMETRIC    FUNCTIONS 


1 

39° 

*i 

29° 

219°        *309^ 

1 

log  sin 

(I. 

log  tan 

c.  d. 

log  cot 

log  cos 

d. 

Prop.  Pts. 

0 

9.79  887 

9.90  837 

0.09   163 

9.89  050 

60 

1 

9.79  903 

i6 

9.90  863 

26 

0.09   137 

9.89  040 

59 

2 

9.79  918 

IS 

9.90  889 

26 

0.09   111 

9.89  030 

58 

3 

9.79  934 

i6 
i6 
IS 

9.90  914 

25 
26 
26 

0.09  086 

9.89  020 

57 

4 

5 

9.79  950 

9.90  940 

0.09  060 

9.89  009 

56 

55 

9.79  965 

9.90  966 

0.09  034 

9.88  999 

6 

9.79  981 

i6 

9.90  992 

26 

0.09  008 

9.88  989 

54 

7 

9.79  996 

IS 

9.91   018 

26 

0.08  982 

9.88  978 

53 

8 

9.80  012 

9.91   043 

2S 
26 
26 

0.08  957 

9.88  968 

52 

9 
10 

9.80  027 

IS 
i6 

9.91   069 

0.08  931 

9.88  958 

51 
50 

" 

9.80  043 

9.91   095 

0.08  905 

9.88  948 

11 

9.80  058 

IS 

9.91   121 

26 

0.08   879 

9.88  937 

49 

6 

2.6 

2.S 

12 

9.80  074 

i6 

9.91   147 

26 

0.08   853 

9.88  927 

48 

7 
8 

3.0 

2.9 

13 

9.80  089 

IS 
i6 
15 

9.91   172 

25 

26 
26 

0  08  828 

9.88  917 

47 

3-9 

3-8 

14 

9.80  105 

9.91   198 

0.08  802 

9.88  906 

46 

10 

4.3 
8.7 
13.0 

4.2 

15 

9.80  120 

9.91   224 

0.08   776 

9.88  896 

45 

30 

8.3 
12.5 

16 

9.80  136 

16 

9.91   250 

26 

0.08   750 

9.88  886 

44 

40 

17.3 

16.7 

17 

9.80  151 

IS 

9.91   276 

26 

0.08  724 

9.88  875 

43 

SO 

21.7 

20.8 

18 

9.80  166 

IS 
16 
IS 

9.91   301 

2S 

0.08  699 

9.88  865 

42 

19 
20 

9.80  182 

9.91   327 

26 

0.08  673 

9.88  855 

41 
40 

9.80  197 

9.91  353. 

0.08  647 

9.88  844 

21 

9.80  213 

16 

9.91  379 

26 

0.08  621 

9.88  834 

39 

22 

9.80  228 

IS 

9.91   404 

25 

0.08  596 

9.88  824 

38 

23 

9.80  244 

9.91  430 

0.08  570 

9.88  813 

37 

24 

25 

9.80  259 

IS 

15 

9.91  456 

26 

0.08  544 

9.88  803 

36 
35 

9.80  274 

9.91   482 

0.08  518 

9.88  793 

26 
27 

9.80  290 
9.80  305 

16 

IS 

9.91   507 
"9.91   533 

25 

26 

0.08  493 
0.08  467 

9.88  782 
9.88  772 

34 
33 

tr 

16 

15 

28 

9.80  320 

IS 

9.91  559 

26 

0.08  441 

9.88  761 

32 

6 

1.6 

i-S 

29 

9.80  336 

IS 

9.91  585 

26 

25 

0.08  415 

9.88  751 

31 
30 

7 
8 
9 

1.9 

2.1 
2.4 

1.8 
2.0 
2.2 

30 

9.80  351 

9.91  610 

0.08  390 

9.88  741 

31 

9.80  366 

IS 

9.91  636 

26 

0.08  364 

9.88  730 

29 

10 

2.7 

2-5 

32 

9.80  382 

16 

9.91  662 

26 

0.08  338 

9.88  720 

28 

20 
30 
40 

5-3 

S.o 
7.5 
10.0 

33 

9.80  397 

IS 

9.91  688 

26 

0.08  312 

9.88  709 

27 

10.7 

34 
35 

9.80  412 

IS 

16 

9.91   713 

25 

26 

0.08  287 

9.88  699 

26 

25 

SO 

13-3 

12.S 

9.80  428 

9.91   739 

0.08  261 

9.88  688 

36 

9.80  443 

IS 

9.91   765 

26 

0.08  235 

9.88  678 

24 

37 

9.80  458 

IS 

9.91   791 

26 

0.08  209 

9.88  668 

23 

38 

9.80  473 

IS 

9.91  816 

25 

0.08   184 

9  88  657 

22 

39 
40 

9.80  489 

IS 

9.91   842 

26 

0.08   158 

9.88  647 

21 
20 

9.80  504 

9.91   868 

0.08   132 

9.88  636 

41 

9.80  519 

IS 

9.91   893 

25 

0.08   107 

9.88  626 

19 

42 

9.80  534 

IS 

9.91   919 

26 

0.08  081 

9.88  615 

18 

43 

9.80  550 

16 

9.91   945 

26 

0.08  055 

9.88  605 

17 

44 
45 

9.80  565 

IS 
IS 

9.91  971 

26 

25 

0.08  029 

9.88  594 

16 

15 

6 

11 

I.I 

lU 

I.O 

9.80  580 

9.91   996 

0.08  004 

9.88  584 

46 

9.80  595 

IS 

9.92  022 

26 

0.07  978 

9.88  573 

14 

7 
8 

1.3 

1.2 

47 

9.80  610 

IS 

9.92  048 

26 

0.07  952 

9.88  563 

13 

9 

1.6 

i-S 

48 

9.80  625 

IS 

9.92  073 

25 

0.07  927 

9.88  552 

12 

10 

1.8 

1.7 

49 
50 

9.80  641 

16 

IS 

9.92  099 

26 

26 

0.07  901 

9.88  542 

11 
10 

20 
30 
40 

3-7 

s.s 

7.3 

3.3 
6.7 

9.80  656 

9.92   125 

0.07  875 

9.88  531 

51 

9.80  671 

IS 

9.92   150 

25 

0.07   850 

9.88  521 

9 

SO 

9.2 

8.3 

52 

9.80  686 

IS 

9.92   176 

26 

0.07   824 

9.88  510 

8 

53 

9.80  701 

IS 

9.92  202 

26 

0.07   798 

9.88  499 

7 

54 

55 

9.80  716 

IS 
IS 

9.92   227 

25 
26 

0.07  773 

9.88  489 

6 

5 

9.80  731 

9.92   253 

0.07  747 

9.88  478 

56 

9.80  746 

IS 

9.92   279 

26 

0.07  721 

9.'88  468 

4 

57 

9.80  762 

16 

9.92  304 

25 

0.07  696 

9.88  457 

3 

58 

9.80  777 

IS 

9.92  330 

26 

0.07  670 

9.88  447 

2 

59 
60 

9.80  792 

IS 
IS 

9.92  356 

26 

25 

0.07  644 

9.88  436 

1 
0 

9.80  807 

9.92  381 

0.07  619 

9.88  425 

log  COS 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

t 

Prop.  Pts. 

*I40° 

230° 

*320° 

50° 

1 

68 


Logarithms  of  the  Trigoxometbic  Functions 


40°          *i3o^ 

220°    *3io° 

log  sin 

d. 

log  tan 

c.  d. 

log  cot 

log  COS 

d. 

Pi'op.  Pts. 

9.80  807 

9.92  381 

26 
26 

0.07  619 

9.88  425 

60 

1 

9.80  822 

IS 

9.92  407 

0.07  593 

9.88  415 

59 

2 

9.80  837 

9.92  433 

2S 
26 

0.07  567 

9.88  404 

58 

3 

9.80  852 

9.92  458 

0.07  542 

9.88  394 

57 

4 

9.80  867 

15 

9.92  484 

26 

0.07  516 

9.88  383 

56 

55 

5 

9.80  882 

9.92  510 

0.07  490 

9.88  372 

6 

9.80  897 

15 
^5 

9.92  535 

2S 
26 
26 

0.07  465 

9.88  362 

54 

7 

9.80  912 

9.92  561 

0.07  439 

9.88  351 

53 

8 

9.80  927 

9.92  587 

2S 
26 

0.07  413 

9.88  340 

52 

9 
10 

9.80  942 

IS 

9.92  612 

0.07  388 

9.88  330 

51 
50 

" 

26  ^K      1 

9.80  957 

9.92  638 

0.07  362 

9.88  319 

11 

9.80  972 

9.92  663 

25 

26 

26 

0.07  337 

9.88  308 

49 

6 

2.6 

2.5 

12 

9.80  987 

IS 
IS 
IS 
IS 

9.92  689 

0.07  311 

9.88  298 

48 

7 
g 

3.0 
3.5 

2.9 

13 

9.81  002 

9.92  715 

2S 
26 

26 

0.07  285 

9.88  287 

47 

9 

3.8 

14 

9.81  017 

9.92  740 

0.07  260 

9.88  276 

46 
45 

10 
20 
30 

4.3 
8.7 
I^.O 

4.2 
8.3 
12.5 

15 

9.81  032 

9.92  766 

0.07  234 

9.88  266 

16 

9.81  047 

14 

9.92  792 

0.07  208 

9.88  255 

44 

40 

17.3 

16.7 

17 

9.81  061 

IS 

9.92  817 

0.07  183 

9.88  244 

43 

50 

21.7 

20.8 

18 

9.81  076 

15 

9.92  843 

2S 
26 

26 

0.07  157 

9.88  234 

42 

19 

9.81  091 

IS 
15 

9.92  868 

0.07  132 

9.88  223 

41 
40 

20 

9.81  106 

9.92  894 

0.07  106 

9.88  212 

21 

9.81  121 

IS 

9.92  920 

0.07  080 

9.88  201 

39 

22 

9.81  136 

15 

9.92  945 

2S 
26 

0.07  055 

9.88  191 

38 

23 

9.81  151 

IS 

9.92  971 

2S 
26 

26 

0.07  029 

9.88  180 

37 

24 

25 

9.81  166 
9.81  180 

14 

IS 

9.92  996 

0.07  004 

9.88  169 

36 
35 

9.93  022 

0.06  978 

9.88  158 

26 

9.81  195 

9.93  048 

0.06  952 

9.88  148 

34 

27 

9.81  210 

15 

9.93  073 

25 
26 
2S 
26 

0.06  927 

9.88  137 

33 

1» 

14 

28 

9.81  225 

IS 

9.93  099 

0.06  901 

9.88  126 

32 

6 

1.5 

1.4 

29 
30 

9.81  240 

14 

IS 

9.93  124 

0.06  876 

9.88  115 

31 
30 

7 
8 
9 

1.8 
2.0 
2.^ 

1.6 
1.9 

2.1 

9.81  254 

9.93  150 

0.06  850 

9.88  105 

31 

9.81  269 

9.93  175 

25 
26 
26 

0.06  825 

9.88  094 

29 

10 

2.5 

2.3 

32 

9.81  284 

9.93  201 

0.06  799 

9.88  083 

28 

20 

?-° 

4-7 

33 

9.81  299 

IS 

9.93  227 

0.06  773 

9.88  072 

27 

40 

lO.O 

9-3 

34 
35 

9.81  314 

14 

IS 
IS 

14 

9.93  252 
9.93  278 

26 

0.06  748 

9.88  061 

26 

25 

SO 

12.5 

II. 7 

9.81  328 

0.06  722 

9.88  051 

36 

9.81  343 

9.93  303 

25 

26 

0.06  697 

9.88  040 

24 

37 

9.81  358 

9.93  329 

0.06  671 

9.88  029 

23 

38 

9.81  372 

9.93  354 

25 

26 
26 

0.06  646 

9.88  018 

22 

39 
40 

9.81  387 

IS 

9.93  380 

0.06  620 

9.88  007 

21 
20 

9.81  402 

9.93  406 

0.06  594 

9.87  996 

41 

9.81  417 

9.93  431 

25 
26 

0.06  569 

9.87  985 

19 

42 

9.81  431 

IS 

9.93  457 

0.06  543 

9.87  975 

18 

43 

9.81  446 

9.93  482 

25 

26 

25 

0.06  518 

9.87  964 

17 

44 

45 

9.81  461 

14 

9.93  508 

0.06  492 

9.87  953 

16 
15 

6 

11 

I.I 

IW 

I.O 

9.81  475 

9.93  533 

0.06  467 

9.87  942 

46 

9.81  490 

IS 

9.93  559 

0.06  441 

9.87  931 

14 

7 
3 

1-3 

i.S 

1.7 

1.2 

47 

9.81  505 

9.93  584 

25 

26 
26 

0.06  416 

9.87  920 

13 

9 

I.S 

48 

9.81  519 

9.93  610 

0.06  390 

9.87  909 

12 

10 

1.8 

1.7 

49 

9.81  534 

9.93  636 

0.06  364 

9.87  898 

11 

20 

3-7 

3.3 

15 

25 

10 

30 
40 

S-S 
7.3 

S.o 
6.7 

50 

9.81  549 

9.93  661 

0.06  339 

9.87  887 

51 

9.81  563 

14 

9.93  687 

26 

0.06  313 

9.87  877 

9 

SO 

9.2 

8.3 

52 

9.81  578 

IS 

9.93  712 

25 

26 

0.06  288 

9.87  866 

8 

53 

9.81  592 

14 

9.93  738 

0.06  262 

9.87  855 

7 

54 

55 
56 

9.81  607 

15 
14 

9.93  763 

25 

26 

0.06  237 

9.87  844 

6 

5' 
4 

9.81  622 
9.81  636 

9.93  789 
9.93  814 

0.06  211 

9.87  833 
9.87  822 

25 

0.06  186 

57 

9.81  651 

IS 

9.93  840 

26 

0.06  160 

9.87  811 

3 

58 

9.81  665 

14 

9.93  865 

25 
26 

25 

0.06  135 

9.87  800 

2 

59 
60 

9.81  680 

IS 

14 

9.93  891 

0.06  109 

9.87  789 

1 
0 

9. SI  694 

9.93  916 

0.06  084 

9.87  778 

log  COS 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*i39° 

229° 

*3i9- 

49° 

fiO 


Logarithms  of  the  Trigonometric  Functions 


41° 

* 

31° 

221^            *^H° 

/ 

log  sin 

d. 

log  tan 

c.  d. 

log  cot 

log  cos 

d. 

Prop.  Pts. 

0 

9.81   694 

9.93  916 

0.06  084 

9.87  778 

60 

1 

9.81   709 

IS 

9.93  942 

26 

0.06  058 

9.87  767 

11 

59 

2 

9.81   723 

1.4 

9.93  967 

25 

0.06  033 

9.87  756 

II 

58 

3 

9.81  738 

IS 

9.93  993 

26 

0.06  007 

9.87  745 

57 

4 

5 

9.81  752 

14 
IS 

9.94  018 

2S 
26 

0.05  982 

9.87  734 

II 
II 

56 

55 

9.81  767 

9.94  044 

0.05  956 

9.87  723 

6 

9.81  781 

14 

9.94  069 

2S 

0.05  931 

9.87  712 

II 

54 

7 

9.81  796 

IS 

9.94  095 

26 

0.05  905 

9.87  701 

II 

53 

8 

9.81  810 

14 

9.94   120 

2S 

0.05  880 

9.87  690 

52 

9 

9.81  825 

IS 

9.94  146 

0.05  854 

9.87  679 

51 

25 

20 

25 

10 

9.81  839 

9.94  171 

0.05   829 

9.87  668 

50 

11 

9.81  854 

IS 

9.94  197 

26 

0.05  803 

9.87  657 

49 

6 

2.6 

2.5 

12 

9.81  868 

14 

9.94  222 

25 

0.05  778 

9.87  646 

48 

7 
g 

3.0 
3.5 

2.9 

13 

9.81  882 

14 

9.94  248 

26 

0.05  752 

9.87  635 

47 

9 

3-8 

14 

9.81  897 

IS 
14 

9.94  273 

25 

26 

0.05  727 

9.87  624 

46 

10 

8.7 

13.0 

4.2 
8.3 
12. s 

15 

9.81   911 

9.94  299 

0.05  701 

9.87  613 

45 

30 

16 

9.81  926 

IS 

9.94  324 

25 

0.05  676 

9.87  601 

44 

40 

17.3 

16.7 

17 

9.81  940 

14 

9.94  350 

26 

0.05  650 

9.87  590 

43 

50 

21.7 

20.8 

18 

9.81   955 

IS 

9.94  375 

25 

0.05  625 

9.87  579 

42 

19 
20 

9.81  969 

14 
14 

9.94  401 

25 

0.05  599 

9.87  568 

41 
40 

9.81  983 

9.94  426 

0.05  574 

9.87  557 

21 

9.81  998 

IS 

9.94  452 

26 

0.05  548 

9.87  546 

39 

22 

9.82  012 

14 

9.94  477 

25 

0.05  523 

9.87  535 

38 

23 

9.82  026 

14 

9.94  503 

26 

0.05  497 

9.87  524 

37 

24 

25 

9.82  041 

IS 
14 

9.94  528 

25 
26 

0.05  472 

9.87  513 

36 

35 

9.82  055 

9.94  554 

0.05  446 

9.87  501 

26 

9.82  069 

14 

9.94  579 

25 

0.05  421 

9.87  490 

34 

27 

9.82  084 

IS 

9.94  604 

25 

0.05  396 

9.87  479 

33 

10 

14 

28 

9.82  098 

14 

9.94  630 

26 

0.05  370 

9.87  468 

32 

6 

1.5 

1.4 

29 

9.82   112 

14 
14 

9.94  655 

25 

26 

0.05  345 

9.87  457 

31 

7 
8 

1.8 

1.6 
1.9 
2.1 

30 

9.82   126 

9.94  681 

0.05  319 

9.87  446 

30 

9 

2.S 

31 

9.82  141 

IS 

9.94  706 

25 

0.05  294 

9.87  434 

29 

10 

2.5 

2.3 

32 

9.82  155 

14 

9.94  732 

26 

0.05  268 

9.87  423 

28 

20 
30 
40 

S-o 
7-S 

lO.O 

4-7 
7.0 
9-3 

33 

9.82  169 

14 

9.94  757 

25 

0.05  243 

9.87  412 

27 

34 
35 

9.82   184 

IS 
14 

9.94  783 

26 

25 

0.05  217 

9.87  401 

26 

25 

SO 

12.5 

II. 7 

9.82   198 

9.94  808 

0.05   192 

9.87  390 

36 

9.82  212 

14 

9.94  834 

26 

0.05   166 

9.87  378 

24 

37 

9.82  226 

14 

9.94  859 

25 

0.05   141 

9.87  367 

23 

38 

9.82  240 

14 

9.94  884 

25 

0.05   116 

9.87  356 

22 

39 

9.82  255 

IS 
14 

9.94  910 

26 

25 

0.05  090 

9.87  345 

21 

40 

9.82  269 

9.94  935 

0.05  065 

9.87  334 

20 

41 

9.82  283 

14 

9.94  961 

26 

0.05  039 

9.87  322 

19 

42 

9.82  297 

14 

9.94  986 

25 

0.05  014 

9.87  311 

18 

43 

9.82  311 

14 

9.95  012 

26 

0.04  988 

9.87  300 

17 

44 

9.82  326 

IS 
14 

9.95  037 

25 

25 

0.04  963 

9.87  288 

16 
15 

6 

12 

1.2 

:'   1 

45 

9.82  340 

9.95   062 

0.04  938 

9.87  277 

46 

9.82  354 

14 

9.95  088 

26 

0.04  912 

9.87  266 

14 

7 
8 

1.4 
1.6 

T  8 

3 

47 

9.82  368 

14 

9.95   113 

25 

0.04  887 

9.87  255 

13 

5 

48 

9.82  382 

14 

9.95   139 

26 

0.04  861 

9.87  243 

12 

10 

2.0 

8 

49 

9.82  396 

14 

9.95   164 

25 
26 

0.04  836 

9.87  232 

11 

20 
30 
40 

4.0 

3 

s 

7 

7 
5 
3 

50 

9.82  410 

9.95   190 

0.04  810 

9.87  221 

10 

8.0 

51 

9.82  424 

14 

9.95  215 

25 

0.04  785 

9.87  209 

9 

SO 

lO.O 

9 

2 

52 

9.82  439 

IS 

9.95  240 

25 

0.04  760 

9.87  198 

8 

53 

9.82  453 

14 

9.95   266 

26 

0.04  734 

9.87  187 

7 

54 
55 

9.82  467 

14 
14 

9.95  291 

25 
26 

0.04  709 

9.87  175 

6 

5 

9.82  481 

9.95  317 

0.04  683 

9.87   164 

56 

9.82  495 

14 

9.95  342 

25 

0.04  658 

9.87  153 

4 

57 

9.82  509 

14 

9.95  368 

26 

0.04  632 

9.87  141 

3 

58 

9.82  523 

14 

9.95  393 

25 

0.04  607 

9.87  130 

2 

59 
60 

9.82  537 

14 
14 

9.95  418 

2S 

26 

0.04  582 

9.87  119 

1 
0 

9.82  551 

9.95  444 

0.04  556 

9.87  107 

log  COS 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*i38°        : 

228° 

*3i8° 

48° 

1 

7(\ 


Logarithms  of  the  Trigoxometric  Functions 


42°                *i32 

°         222°        *3I2° 

"o 

log  sin 

d. 

log  tan 

c.  d. 

log  cot 

log  cos 

d. 

Prop.  Pts. 

9.82  551 

9.95  444 

0.04  556 

9.87  107 

60 

1 

9.82  565 

9.95  469 

26 

0.04  531 

9.87  096 

^^ 

59 

2 

9.82  579 

14 

9.95   495 

25 

25 

0.04  505 

9.87  085 

58 

3 

9.82  593 

9.95   520 

0.04  480 

9.87  073 

57 

4 

9.82  607 

14 

9.95   545 

26 

.0.04  455 

9.87  062 

12 

56 

55 

5 

9.82  621 

9.95   571 

0.04  429 

9.87  050 

6 

9.82  635 

14 

9.95   596 

26 

0.04  404 

9.87  039 

54 

7 

9.82  649 

14 

9.95  622 

25 

0.04  378 

9.87  028 

53 

8 

9.82  663 

9.95  647 

0.04  353 

9.87  016 

52 

9 
10 

9.82  677 

14 

9.95  672 

26 

0.04  328 

9.87  005 

12 

51 
60 

6 

7 
8 

26 

2.6 
3-0 

3-5 

26 

9.82  691 

9.95   698 

0.04  302 

9.86  993 

11 
12 

9.82  705 
9.82  719 

14 

14 
14 
14 
14 

9.95   723 
9.95   748 

25 
25 

26 

0.04   277 
0.04  252 

9.86  982 
9.86  970 

12 

49 

48 

2-S 
2.9 
3-3 

13 

9.82  733 

9.95   774 

25 
26 

0.04   226 

9.86  959 

47 

9 

3-9 

3.8 

14 

9.82  747 

9.95   799 

0.04  201 

9.86  947 

II 

46 

20 
30 

4-3 
8.7 
13  0 

4.2 
8.3 
12.5 

15 

9.82  761 

9.95   825 

0.04   175 

9.86  936 

16 

9.82  775 

9.95  850 

25 

0.04   150 

9.86  924 

44 

40 
5° 

17-3 
21.7 

16.7 
20.8 

17 

9.82  788 

14 
14 
14 

9.95  875 

26 

0.04   125 

9.86  913 

43 

IS 

9.82  802 

9.95  901 

25 
26 

0.04  099 

9.86  902 

42 

19 
20 

9.82  816 

9.95   926 

0.04  074 

9.86  890 

II 

41 
40 

9.82  830 

9.95  952 

0.04  048 

9.86  879 

21 

9.82  844 

14 

9.95  977 

25 

0.04  023 

9.86  867 

39 

22 

9.82  858 

14 
14 

9.96  002 

26 

0.03  998 

9.86  855 

12 

38 

23 

9.82  872 

9.96  028 

25 

25 

26 

0.03  972 

9.86  844 

37 

24 

9.82  885 

9.96  053 

0.03  947 

9.86  832 

II 

36 
35 

25 

9.82  899 

9.96  078 

0.03  922 

9.86  821 

26 

9.82  913 

14 

9.96  104 

0.03  896 

9.86  809 

34 

// 

14    13              1 

27 

9.82  927 

14 
14 
13 

9.96  129 

26 

25 
25 

26 

0.03   871 

9.86  798 

II 

33 

28 

9.82  941 

9.96  155 

0.03   845 

9.86  786 

32 

6 

1-4 

1-3 

29 

9.82  955 

9.96  180 

0.03  820 

9.86  775 

12 

31 
30 

7 
8 
9 

I.O 
1.9 

2.1 

1-5 
1.7 
2.0 

30 

9.82  968 

9.96  205 

0.03   795 

9.86  763 

31 

9.82  982 

9.96  231 

0.03   769 

9.86  752 

29 

10 

2.3 

2.2 

32 

9.82  996 

14 
13 
14 

9.96  256 

0.03   744 

9.86  740 

28 

30 

7.0 

6.5 

33 

9.83  010 

9.96  281 

26 

0.03  719 

9.86  728 

27 

40 

9-3 

8.7 

34 

9.83  023 

9.96  307 

25 

0.03  693 

9.86  717 

12 

26 

25 

SO 

II. 7  10.8               1 

35 

9.83  037 

9.96  332 

0.03  668 

9.86  705 

36 

9.83  051 

14 

9.96  357 

25 

26 

0.03  643 

9.86  694 

24 

37 

9.83  065 

9.96  383 

0.03  617 

9.86  682 

23 

38 

9.83  078 

14 
14 

9.96  408 

25 
26 

0.03   592 

9.86  670 

12 

22 

39 

9.83  092 

9.96  433 

0.03   567 

9.86  659 

12 

21 
20 

40 

9.83   106 

9.96  459 

0.03   541 

9.86  647 

41 

9.83   120 

14 

9.96  484 

25 
26 

0.03   516 

9.86  635 

19 

42 

9.83   133 

13 

9.96  510 

0.03   490 

9.86  624 

18 

43 

44 

9.83   147 
9.83   161 

14 
13 

9.96  535 
9.96  560 

25 

26 

0.03  465 
0.03   440 

9.86  612 
9.86  600 

12 
II 

17 
16 
15 

// 

6 

12 

1.2 

11 

I.I 

45 

9.83   174 

9.96  586 

0.03  414 

9.86  589 

46 

9.83   188 

14 

9.96  611 

25 

0.03   389 

9.86  577 

14 

7 
8 

1.4 
T  6 

1-3 
i.S 

47 

9.83  202 

14 

9.96  636 

25 

26 

0.03  364 

9.86  565 

13 

9 

1.8 

1-6 

48 

9.83  215 

13 

9.96  662 

0.03  338 

9.86  554 

12 

10 

2.0 

1.8 

49 

9.83  229 

13 

9.96  687 

25 

0.03  313 

9.86  542 

12 

11 
10 

30 
40 

6.0 
8.0 

3-7 

5-5 
7-3 

50 

9.83  242 

9.96  712 

0.03  288 

9.86  530 

51 

9.83  256 

14 

9.96  738 

26 

0.03   262 

9.86  518 

12 

9 

SO 

52 

9.83  270 

14 

9.96  763 

25 

0.03   237 

9.86  507 

8 

53 

9.83  283 

13 

9.96  788 

25 

26 
25 

0.03   212 

9.86  495 

7 

54 
55 

9.83  297 

14 
13 

9.96  814 

0.03   186 

9.86  483 

II 

6 

5 

9.83  310 

9.96  839 

0,03   161 

9.86  472 

56 

9.83  324 

14 

9.96  864 

25 

0  03   136 

9.86  460 

12 

4 

57 

9.83  338 

14 

9.96  890 

0.03   110 

9.86  448 

3 

58 

9.83  351 

13 

9.96  915 

25 

0.03  085 

9.86  436 

2 

59 
60 

9.83  365 

14 
13 

9.96  940 

25 

26 

0.03  060 

9.86  425 

12 

1 
0 

9.83  378 

9.96  966 

0.03  034 

9.86  413 

log  COS 

d. 

log  cot 

c.  d. 

log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*i37° 

227° 

*3i7" 

470 

71 


Logarithms  of  the  Trigonometric  Functions 


43° 

* 

[33' 

223°        *3i3° 

1 

log  sin 

d. 

log  tan 

c.  d. 

log  cot 

log  cos 

d. 

Prop.  Pts. 

9.83  378 

9.96  966 

0.03  034 

9.86  413 

60 

1 

9.83  392 

9.96  991 

25 
26 

0.03  009 

9.86  401 

59 

2 

9.83  405 

14 
13 
14 

9.97  016 

0.02  984 

9.86  389 

58 

3 

9.83  419 

9.97  042 

25 
25 

26 

0.02  958 

9.86  377 

57 

4 

9.83  432 

9.97  067 

0.02  933 

9.86  366 

12 

56 

55 

5 

9.83  446 

9.97  092 

0.02  908 

9.86  354 

6 

9.83  459 

13 

9.97   118 

0.02  882 

9.86  342 

54 

7 

9.83  473 

9.97   143 

25 
25 
26 

0.02  857 

9.86  330 

12 

53 

8 

9.83  486 

14 
13 

9.97   168 

0.02   832 

9.86  318 

52 

9 
10 

9.83  500 

9.97   193 

0.02   807 

9.86  306 

II 

51 
50 

// 

26 

25 

9.83  513 

9.97   219 

0.02   781 

9.86  295 

11 

9.83  527 

14 

9.97  244 

25 

0.02   756 

9.86  283 

49 

7 

8 

2.5 

12 

9.83  540 

9.97   269 

26 

0.02   731 

9.86  271 

48 

S-5 

i-i 

13 

9.83  554 

13 
14 

9.97  295 

25 

25 

0.02  705 

9.86  259 

47 

9 

3.g 

3.8 

14 

9.83  567 

9.97  320 

0.02   680 

9.86  247 

12 

46 
45 

10 
20 
30 

4-3 
8.7 
13.0 

4.2 
8.3 
12. S 

15 

9.83  581 

9.97  345 

0.02  655 

9.86  235 

16 

9.83  594 

13 

9.97  371 

0.02   629 

9.86  223 

44 

40 

17-3 

16.7 

17 

9.83  608 

14 

9.97  396 

25 

0.02  604 

9.86  211 

43 

18 

9.83  621 

9.97  421 

0.02   579 

9.86  200 

42 

19 

9.83  634 

14 

9.97  447 

25 

0.02   553 

9.86  188 

12 

41 
40 

20 

9.83  648 

9.97  472 

0.02  528 

9.86  176 

21 

9.83  661 

13 

9.97  497 

25 
26 

0.02   503 

9.86  164 

39 

22 

9.83  674 

13 

9.97  523 

0.02  477 

9.86  152 

38 

23 

9.83  688 

14 

9.97  548 

25 

25 
25 

26 

0.02  452 

9.86  140 

12 

37 

24 

9.83  701 

14 

9.97  573 

0.02   427 

9.86   128 

12 

36 

35 

25 

9.83  715 

9.97  598 

0.02  402 

9.86   116 

26 

9.83  728 

9.97  624 

0.02  376 

9.86  104 

34 

27 

9.83  741 

9.97  649 

25 

0.02  351 

9.86  092 

33 

" 

14 

IS 

28 

9.83  755 

13 
13 

9.97  674 

25 
26 

0.02  326 

9.86  080 

32 

6 

1.4 

1-3 

29 

9.83  768 

9.97  700 

25 

0.02  300 

9.86  068 

12 

31 
30 

7 
8 
9 

1.6 
1.9 
2.1 

i-S 
1-7 
2.0 

30 

9.83  781 

9.97  725 

0.02  275 

9.86  056 

31 

9.83  795 

14 

9.97  750 

25 

26 

0.02  250 

9.86  044 

29 

10 

2.3 

2.2 

32 

9.83  808 

13 

9.97  776 

0.02   224 

9.86  032 

28 

20 

4-7 

4-3 
6.5 
8.7 

33 

9.83  821 

9.97  801 

25 
25 

0.02   199 

9.86  020 

27 

40 

9-3 

34 

9.83  834 

14 

9.97  826 

0.02   174 

9.86  008 

12 

26 

25 

SO 

11.7 

10.8 

35 

9.83  848 

9.97  851 

0.02   149 

9.85  996 

36 

9.83  861 

13 

9.97  877 

0.02   123 

9.85  984 

24 

37 

9.83  874 

9.97  902 

25 

0.02   098 

9.85  972 

23 

38 

9.83  887 

9.97  927 

26 

25 

0.02   073 

9.85  960 

22 

39 

9.83  901 

13 

9.97  953 

0.02  047 

9.85  948 

12 

21 
20 

40 

9.83  914 

9.97  978 

0.02  022 

9.85  936 

41 

9.83  927 

13 

9.98  003 

25 
26 

0.01   997 

9.85  924 

19 

42 

9.83  940 

13 

9.98  029 

0.01   971 

9.85  912 

18 

43 

9.83  954 

9.98  054 

0.01   946 

9.85  900 

17 

44 
45 

9.83  967 

13 

9.98  079 

25 

0.01   921 

9.85  888 

12 

16 
15 

n 
6 

12 

1.2 

11 

I.I 

9.83  980 

9.98   104 

0.01   896 

9.85  876 

46 

9.83  993 

13 

9.98  130 

26 

0.01   870 

9.85  864 

12 

14 

7 

1.4 

1-3 

47 

9.84  006 

13 

9. 98   155 

25 

0.01   845 

9.85  851 

13 

13 

8    i.O 

i-S 
1-6 
1.8 

48 

9.84  020 

9.98  180 

25 
26 

0.01    820 

9.85  839 

12 

10 

2.0 

49 

9.84  033 

9.98  206 

0.01    794 

9.85  827 

11 

20 

4.0 

3-7 

50 

9.84  046 

25 

10 

30 

40 

6.0 
8.0 

5-5 
7-3 

9.98  231 

0.01   769 

9.85  815 

51 

9.84  059 

13 

9.98  256 

25 

0.01    744 

9.85  803 

12 

9 

50 

lO.O 

9.2 

52 

9.84  072 

13 

9.98  281 

25 

0.01   719 

9.85  791 

12 

8 

53 

9.84  085 

9.98  307 

0.01   693 

9.85  779 

7 

54 

9.84  098 

14 

9.98  332 

25 

0.01   668 

9.85  766 

12 

6 

5 

55 

9.84  112 

9.98  357 

0.01   643 

9.85  754 

56 

9.84   125 

13 

9.98  383 

26 

0.01   617 

9.85  742 

12 

4 

57 

9.84  138 

13 

9.98  408 

25 

0.01   592 

9.85  730 

12 

3 

58 

9.84   151 

13 

9.98  433 

25 

0.01   567 

9.85  718 

12 

2 

59 

9.84  164 

13 

9.98  458 

25 

26 

0.01   542 

9.85  706 

13 

1 
0 

60 

9.84   177 

9.98  484 

0.01   516 

9.85  693 

log  COS 

d. 

log  cot 

c.  d. 

.  log  tan 

log  sin 

d. 

/ 

Prop.  Pts. 

*I36°        . 

226° 

*3i6° 

46° 

72 


Logarithms  of  the  Trigonometric  Functions 


^50 


^134^ 


224" 


*3i4' 


log  sill   i    (1. 


log  tan   c.  (1.    log  cot      log  cos      d. 


Prop.  Pts. 


0 
1 

2 
3 

_4 
5 
6 
7 
8 

_9^ 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
58 

40 

41 
42 
43 
44 

"45" 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 

i^ 
60 


9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9.84 

9^84_ 

9.84 


177 
190 
203 
216 
229 
242 
255 
269 
282 
295 
308 
321 
334 
347 
360 
373 
385 
398 
411 
424 
437 
450 
463 
476 
489 
502 
515 
528 
540 
553 
566 
579 
592 
605 
618 
630 
643 
656 
669 
682 
"694" 
707 
720 
733 
745 
758 
771 
784 
796 
809 
822 
835 
847 
860 
873 
885 
898 
911 
923 
936 
949 


log  cos 


9.98  484 
9.98  509 
9.98  534 
9.98  560 
9.98  585 


9.98  610 
9.98  635 
9.98  661 
9.98  686 
9.98  711 


9.98  863 
9.98  888 
9.98  913 
9.98  939 
9.98  964 


9.98  989 

9.99  015 
9.99  040 
9.99  065 
9.99  090 


9.99 
9.99 
9.99 
9.99 


116 
141 
166 
191 


9.99  217 


9.99  242 
9.99  267 
9.99  293 
9.99  318 
9.99  343 


9.99  368 
9.99  394 
9.99  419 
9.99  444 
9.99  469 


9.99  495 
9.99  520 
9.99  545 
9.99  570 
9.99  596 


9.99  621 
9.99  646 
9.99  672 
9.99  697 
9.99  722 


9.99  747 
9.99  773 
9.99  798 
9.99  823 
9.99  848 


^135° 


225° 


9.99  874 
9.99  899 
9.99  924 
9.99  949 
9.99  975 
0.00  000 

log  cot 


^315^ 


25 

25 

26 
25 
25 

25 

26 
25 

25 

26 

25 

25 
25 

26 

25 

25 

25 

26 
25 
25 

26 
25 

25 

25 
26 

25 
25 

25 

26 
25 

25 
26 
25 
25 
25 

26 
25 

25 

25 
26 

25 

25 
25 
26 
25 

25 
26 
25 

25 

25 

26 
25 

25 

25 
26 


25 
26 
25 

cTT 


01  516 
01  491 
01  466 
01  440 
01  415 


9.85 
9.85 
9.85 


693 
681 
669 


9.85  657 
9.85  645 


01  390 
01  365 
01  339 
01  314 
01  289 


9.85  632 

9.85  620 

9.85  608 

9.85  596 

9.85  583 


01  263 

01  238 

01  213 

01  188 

01  162 


9.85  571 

9.85  559 

9.85  547 

9.85  534 

9.85  522 


01  137 
01  112 
01  087 
01  061 
01  036 


9.85 
9.85 


510 
497 


9.85  485 
9.85  473 
9.85  460 


01  Oil 
00  985 
00  960 
00  935 
00  910 


9.85 
9.85 


448 
436 


9.85  423 
9.85  411 
9.85  399 


00  884 

00  859 

00  834 

00  809 

00  783 


9.85  386 

9.85  374 

9.85  361 

9.85  349 

9.85  337 


00  758 

00  733 

00  707 

00  682 

00  657 


9.85  324 
9.85  312 


9.85 
9.85 


299 

287 


9.85  274 


00  632 
00  606 
00  581 
00  556 
00  531 


9.85  262 
9.25  250 
9.85  237 
9.85  225 
9.85  212 


00  505 
00  480 
00  455 
00  430 
00  404 


9.85  200 
9.85  187 


9.85 
9.85 
9.85 


175 
162 
150 


00  379 
00  354 
00  328 
00  303 
00  278 


9.85  137 
9.85  125 
9.85  112 
9.85  100 
9.85  087 


00  253 

00  227 

00  202 

00  177 

00  152 


9.85  074 

9.85  062 

9.85  049 

9.85  037 

9.85  024 


00  126 

00  101 

00  076 

00  051 

00  025 

00  000 

log  tan 


9.85  012 
9.84  999 
9.84  986 
9.84  974 
9.84  961 
9.84  949 

log  sin 


73_ 


d. 


26  I  26  I  14 

2.6!  2.5!  1.4 
3.0J  2.9    1.6 

3.51   3-3  1-9 

3-9    3.8  2.1 

4.3    4.2  2.3 

8.7:   8.3  4-7 

13.0,12.5,  7.0 

17.3  16.7:  9-3 
21.7I20.8111.7 


18 

12 

1-3 

1.2 

I-S 
1-7 

i   2.0 

1.4 
1.6 
1.8 

2.2 

2.0 

4-3 
6.5 

8.7 

4.0 
6.0 
8.0 

10.8 

lO.O 

Prop.  Pts. 


TABLE  III 


NATURAL  TRIGONOMETRIC  FUNCTIONS 


FOUR  PLACES 


75 


Natural  Si?rEs  and  Cosines 


/ 

0 

o 

1 

= 

2 

^ 

3 

:> 

4 

/ 

sin 

COS 

sin 

cos 

sin 

cos 

sin 

COS 

sin 

COS 

0 

.0000 

1.000 

.0175 

.9998 

.0349 

.9994 

.0523 

.9986 

.0698 

.9976 

60 

1 

03 

.000 

77 

98 

52 

94 

26 

86 

0700 

75 

59 

2 

06 

.000 

80 

98 

55 

94 

29 

86 

03 

75 

58 

3 

09 

.000 

83 

98 

58 

94 

32 

86 

06 

75 

57 

4 

12 

.000 

86 

98 

61 

93 

3i 

86 

09 

75 

56 

5 

.0015 

1.000 

.0189 

.9998 

.0364 

.9993 

.0538 

.9986 

.0712 

.9975 

55 

6 

17 

.000 

92 

98 

66 

93 

41 

85 

15 

74 

54 

7 

20 

.000 

95 

98 

69 

93 

44 

85 

18 

74 

53 

8 

23 

.000 

98 

98 

72 

93 

47 

85 

21 

74 

52 

9 

26 

.000 

0201 

98 

75 

93 

50 

8i 

24 

74 

51 

10 

.0029 

1.000 

.0204 

.9998 

.0378 

.9993 

.0552 

.9985: 

.0727 

.9974 

50 

11 

32 

.000 

07 

98 

81 

93 

55' 

85 

29 

73 

49 

12 

35 

.000 

09 

98 

84 

93 

58 

84 

32 

73 

48 

13 

38 

.000 

12 

98 

87 

93 

61 

84 

35 

73 

47 

14 

41 

.000 

15 

98 

90 

92 

64 

84 

38 

73 

46 

15 

.0044 

1.000 

.0218 

.9998 

.0393 

.9992 

.0567 

.9984 

.0741 

.9973 

45 

16 

47 

.000 

21 

98 

96 

92 

70 

84 

44 

72 

44 

17 

49 

.000 

24 

97 

98 

92 

73 

84 

47 

72 

43 

18 

52 

.000 

27 

97 

0401 

92 

76 

83 

50 

72 

42 

19 

55 

.000 

30 

97 

04 

92 

79 

83 

53 

72 

41 

20 

.0058 

1.000 

.0233 

.9997 

.0407 

.9992 

.0581 

.9983 

.0756 

.9971 

40 

21 

6] 

.000 

36 

97 

10 

92 

84 

83 

58 

71 

39 

22 

64 

.000 

39 

97 

13 

91 

87 

83 

61 

71 

38 

23 

67 

.000 

41 

97 

16 

91 

90 

83 

64 

71 

37 

24 

70 

.000 

44 

97 

19 

91 

93 

82 

67 

71 

36 

25 

.0073 

1.000 

.0247 

.9997 

.0422 

.9991 

.0596 

.9982 

.0770 

.9970 

35 

26 

76 

.000 

50 

97 

25 

91 

99 

82 

73 

70 

34 

27 

79 

.000 

53 

97 

27 

91 

0602 

82 

76 

70 

33 

28 

81 

.000 

56 

97 

30 

91 

05 

82 

79 

70 

Zl 

29 

84 

.000 

59 

97 

33 

91 

08 

82 

82 

69 

31 

30 

.0087 

1.000 

.0262 

.9997 

.0436 

.9990 

.0610 

.9981 

.0785 

.9969 

30 

31 

90 

.000 

65 

96 

39 

90 

13 

81 

87 

69 

29 

32 

93 

.000 

68 

96 

42 

90 

16 

81 

90 

69 

28 

33 

96 

.000 

70 

% 

45 

90 

19 

81 

93 

68 

27 

34 

99 

.000 

73 

96 

48 

90 

22 

81 

% 

68 

26 

35 

.0102 

.9999 

.0276 

.9996 

.0451 

.9990 

.0625 

.9980 

.0799 

.9968 

25 

36 

05 

99 

79 

96 

54 

90 

28 

80 

0802 

68 

24 

37 

08 

99 

82 

96 

57 

90 

31 

80 

05 

68 

23 

38 

11 

99 

85 

96 

59 

89 

34 

80 

08 

67 

22 

39 

13 

99 

88 

% 

62 

89 

37 

80 

11 

67 

21 

40 

.0116 

.9999 

.0291 

.9996 

.0465 

.9989 

.0640 

.9980 

.0814 

.9967 

20 

41 

19 

99 

94 

96 

68 

89 

42 

79 

16 

67 

19 

42 

22 

99 

97 

96 

71 

89 

45 

79 

19 

66 

IS 

43 

25 

99 

0300 

96 

74 

89 

48 

79 

22 

66 

17 

44 

28 

99 

02 

95 

77 

89 

51 

79 

25 

66 

16 

45 

.0131 

.9999 

.0305 

.9995 

.0480 

.9988 

.0654 

.9979 

.0828 

.9966 

15 

46 

34 

99 

.  08 

95 

83 

88 

57 

78 

31 

65 

14 

47 

37 

99 

11 

95 

86 

88 

60 

78 

34 

65 

13 

48 

40 

99 

14 

95 

88 

88 

63 

78 

37 

65 

12 

49 

43 

99 

17 

95 

91 

88 

66 

78 

40 

65 

11 

50 

.0145 

.9999 

.0320 

.9995 

.0494 

.9988 

.0669 

.9978 

.0843 

.9964 

10 

51 

48 

99 

23 

95 

97 

88 

71 

77 

45 

64 

9 

52 

51 

99 

26 

95 

0500 

87 

74 

77 

48 

64 

8 

53 

54 

99 

29 

95 

03 

87 

77 

77 

51 

64 

7 

54 

57 

99 

32 

95 

06 

87 

80 

77 

54 

63 

6 

55 

.0160 

.9999 

.0334 

.9994 

.0509 

.9987 

.0683 

.9977 

.0857 

.9963 

5 

56 

63 

99 

37 

94 

12 

87 

86 

76 

60 

63 

4 

57 

66 

99 

40 

94 

15 

87 

89 

76 

63 

63 

3 

58 

69 

99 

43 

94 

18 

87 

92 

76 

66 

62 

2 

59 

72 

99 

46 

94 

20 

86 

95 

76 

69 

62 

1 

60 

.0175 

.9998 

.0349 

.9994 

.0523 

.9986 

.0698 

.9976 

.0872 

.9962 

0 

cos 

sin 

cos 

sin 

cos 

gin 

cos 

sin 

COS 

sin 

/ 

89° 

88° 

87° 

86° 

85° 

/ 

76 


Natural  Tangents  and  Cotangents 


r 

0° 

1^ 

2° 

3° 

4° 

/ 

tan   cot 

tan   cot 

tan   cot 

tan 

cot 

tan 

cot 

0 

.0000  Infinite 

.0175  57.2900 

.0349  28.6363 

.0524 

19.0811 

.0699 

14.3007 

60 

1 

03  3437.75 

77  56.3506 

52   3994 

27 

18.9755 

0702 

2411 

59 

2 

06  1718.87 

80  55.4415 

55    1664 

30 

8711 

05 

1821 

58 

3 

09  1145.92 

83  54.5613 

58  27.9372 

33 

7678 

08 

1235 

57 

4 

12  859.436 

86  53.7086 

61   7117 

36 

6656 

11 

0655 

56 

5 

.0015  687.549 

.0189  52.8821 

.0364  27.4899 

.0539 

18.5645 

.0714 

14.0079 

55 

6 

17  572.957 

92   0807 

67   2715 

42 

4645 

17 

13.9507 

54 

7 

20  491.106 

95  51.3032 

70   0566 

44 

3655 

20 

8940 

53 

8 

23  429.718 

98  50.5485 

73  26.8450 

47 

2677 

23 

8378 

52 

9 

26  381.97] 

0201  49.  Si:  7 

75   6367 

50 

1708 

26 

7821 

51 

10 

.0029  343.774 

.0204  49.1039 

.0378  26.4316 

.0553 

18.0750 

.0729 

13.7267 

50 

11 

32  312.521 

07  48.4121 

81   2296 

56 

17.9802 

31 

6719 

49 

12 

35  286.478 

09  47.7395 

84   0307 

59 

8863 

34 

6174 

48 

13 

38  264.441 

12   0853 

87  25.8348 

62 

7934 

37 

5634 

47 

14 

41  245.552 

15  46.4489 

90   6418 

65 

7015 

40 

5098 

46 

15 

.0044  229.182 

.0218  45.8294 

.0393  25.4517 

.0568 

17.6106 

.0743 

13.4566 

45 

16 

47  214.858 

21   2261 

96   2644 

71 

5205 

46 

4039 

44 

17 

49  202.219 

24  44.6386 

99   0798 

74 

4314 

49 

3515 

43 

IS 

52  190.984 

27   0661 

0402  24.8978 

77 

3432 

52 

2996 

42 

19 

55  180.932 

30  43.5081 

05    7185 

80 

2558 

55 

2480 

41 

20 

.0058  171.885 

.0233  42.9641 

.0407  24.5418 

.0582 

17.1693 

.0758 

13.1969 

40 

21 

61  163.700 

36   4335 

10   3675 

85 

0837 

61 

1461 

39 

22 

64  156.259 

39  41.9158 

13   1957 

88 

16.9990 

64 

0958 

38 

23 

67  149.465 

41   4106 

16   0263 

91 

9150 

67 

0458 

37 

24 

70  143.237 

44  40.9174 

19  23.8593 

94 

8319 

69 

12.9962 

36 

25 

.0073  137.507 

.0247  40.4358 

.(H22  23.6945 

.0597 

16.7496 

.0772 

12.9469 

35 

26 

76  132.219 

50  39.9655 

25   5321 

0600 

6681 

75 

8981 

34 

27 

79  127.321 

53   5059 

28   3718 

03 

5874 

78 

8496 

33 

28 

81  122.774 

56   0568 

31   2137 

06 

5075 

81 

8014 

31 

29 

84  118.540 

59  38.6177 

34   0577 

09 

4283 

84 

7536 

31 

30 

.0087  114.589 

.0262  38.1885 

.0437  22.9038 

.0612 

16.3499 

.0787 

12.7062 

30 

31 

90  110.892 

65  37.7686 

40   7519 

15 

2722 

90 

6591 

29 

32 

93  107.426 

68   3579 

42   6020 

17 

1952 

93 

6124 

28 

33 

96  104.171 

71  36.9560 

45   4541 

20 

1190 

96 

5660 

27 

34 

99  101.107 

74   5627 

48   3081 

23 

0435 

99 

5199 

26 

35 

.0102  98.2179 

.0276  36.1776 

.0451  22.1640 

.0626 

15.9687 

.0802 

12.4742 

25 

36 

05  95.4895 

79  35.8006 

54   0217 

29 

8945 

05 

4288 

24 

37 

08  92.9085 

82   4313 

57  21.8813 

32 

8211 

08 

3838 

23 

38 

11  90.4633 

85   0695 

60   7426 

35 

7483 

10 

3390 

22 

39 

13  88.1436 

88  34.7151 

63   6056 

38 

6762 

13 

2946 

21 

40 

.0116  85.9398 

.0291  34.3678 

.0466  21.4704 

.'0641 

15.6048 

.0816 

12.2505 

20 

41 

19  83.8435 

94   0273 

•  69   3369 

44 

5340 

19 

2067 

19 

42 

22  81.8470 

97  33.6935 

72   2049 

47 

4638 

22 

1632 

18 

43 

25  79.9434 

0300   3662 

75   0747 

iO 

3943 

25 

1201 

17 

44 

28  78.1263 

03   0452 

77  20.9460 

53 

3254 

28 

0772 

16 

45 

.0131  76.3900 

.0306  32.7303 

.0480  20.8188 

.0655 

15.2571 

.0831 

12.0346 

15 

46 

34  74.7292 

OS   4213 

83   6932 

58 

1893 

34 

11.9923 

14 

47 

37  73.1390 

11    1181 

86   5691 

61 

1222 

37 

9504 

13 

48 

40  71.6151 

14  31.8205 

89   4465 

64 

0557 

40 

9087 

12 

49 

43  70.1533 

17   5284 

92   3253 

67 

14.9898 

43 

8673 

11 

50 

.0145  68.7501 

.0320  31.2416 

.0495  20.2056 

.0670- 

14.9244 

.0846 

11.8262 

10 

51 

48  67.4019 

23  30.9599 

98   0872 

73 

8596 

49 

7853 

9 

52 

51  66.1055 

26   6833 

0501  19.9702 

76 

7954 

51 

7448 

8 

53 

54  64.8580 

29   4116 

04   85H6 

79 

7317 

54 

7045 

7 

54 

57  63.6567 

32   1446 

07   7403 

82 

6685 

57 

6645 

6 

55 

.0160  62.4992 

.0335  29.8823 

.0509  19.6273 

.0685 

14.6059 

.0860 

11.6248 

5 

56 

63  61.3829 

38   6245 

12   5156 

88 

5438 

63 

5853 

4 

57 

66  60.3058 

40   3711 

15   4051 

90 

4823 

66 

5461 

3 

58 

69  59.2659 

43   1220 

18   J959 

93 

4212 

69 

5072 

2 

59 

72  58.2612 

46  28.8771 

21   1879 

96 

3607 

72 

4685 

1 

60 

.0175  57.2900 

.0349  28.6363 

.0524  19.0811 

.0699 

14.3007 

.0875 

11.4301 

0 

/ 

cot   tan 

cot   tan 

cot   tan 

cot 

tan 

cot 

tan 

89^ 

88^ 

87= 

86° 

85^ 

/ 

77 


Natural  Sines  and  Cosines 


/ 

6 

" 

6 

o 

7 

o 

8 

o 

9 

3 

1 

sin 

cos 

sin 

COS 

sin 

COS 

sin 

COS 

sin 

COS 

0 

.0872 

.9962 

.1045 

.9945 

.1219 

.9925 

.1392 

.9903 

.1564 

.9877 

60 

1 

74 

62 

48 

45 

22 

25 

95 

02 

67 

76 

59 

2 

77 

61 

51 

45 

24 

25 

97 

02 

70 

76 

58 

3 

80 

61 

54 

44 

27 

24 

1400 

01 

73 

76 

57 

4 

83 

61 

57 

44 

30 

24 

03 

01 

76 

75 

56 

6 

.0886 

.9961 

.1060 

.9944 

.1233 

.9924 

.1406 

.9901 

.1579 

.9875 

55 

6 

89 

60 

63 

43 

36 

23 

09 

00 

82 

74 

54 

7 

92 

60 

66 

43 

39 

23 

12 

00 

84 

74 

53 

8 

95 

60 

68 

43 

42 

23 

15 

9899 

87 

73 

52 

9 

98 

60 

71 

42 

45 

22 

18 

99 

90 

73 

51 

10 

.0901 

.9959 

.1074 

.9942 

.1248 

.9922 

.1421 

.9899 

.1593 

.9872 

50 

11 

03 

59 

77 

42 

50 

22 

23 

98 

96 

72 

49 

12 

06 

59 

80 

42 

53 

21 

26 

98 

99 

71 

48 

13 

09 

59 

83 

41 

56 

21 

29 

97 

1602 

71 

47 

14 

12 

58 

86 

41 

59 

20 

32 

97 

05 

70 

46 

15 

.0915 

.9958 

.1089 

.9941 

.1262 

.9920 

.1435 

.9897 

.1607 

.9870 

45 

16 

18 

58 

92 

40 

65 

20 

38 

96 

10 

69 

44 

17 

21 

58 

94 

40 

68 

19 

41 

96 

13 

69 

43 

18 

24 

57 

97 

40 

71 

19 

44 

95 

16 

69 

42 

19 

27 

57 

1100 

39 

74 

19 

46 

95 

19 

68 

41 

20 

.0929 

.9957 

.1103 

.9939 

.1276 

.9918 

.1449 

.9894 

.1622 

.9868 

40 

21 

32 

56 

06 

39 

79 

18 

52 

94 

25 

67 

39 

22 

35 

56 

09 

38 

82 

17 

55 

94 

28 

67 

38 

23 

38 

56 

12 

38 

85 

17 

58 

93 

30 

66 

37 

24 

41 

56 

15 

38 

88 

17 

61 

93 

33 

66 

36 

25 

.0944 

.9955 

.1118 

.9937 

.1291 

.9916 

.1464 

.9892 

.1636 

.9865 

35 

26 

47 

55 

20 

37 

94 

16 

67 

92 

39 

65 

34 

27 

50 

55 

23 

37 

97 

16 

69 

91 

42 

64 

33 

28 

53 

55 

26 

36 

99 

15 

72 

91 

45 

64 

32 

29 

56 

54 

29 

36 

1302 

15 

75 

91 

48 

63 

31 

30 

.0958 

.9954 

.1132 

.9936 

.1305 

.9914 

.1478 

.9890 

.1650 

.9863 

30 

31 

61 

54 

35 

35 

08 

14 

81 

90 

53 

62 

29 

32 

64 

53 

38 

35 

11 

14 

84 

89 

56 

62 

28 

33 

67 

53 

41 

35 

14 

13 

87 

89 

59 

61 

27 

34 

70 

53 

44 

34 

17 

13 

90 

88 

62 

61 

26 

35 

.0973 

.9953 

.1146 

.9934 

.1320 

.9913 

.1492 

.9888 

.1665 

.9860 

26 

36 

76 

52 

49 

34 

23 

12 

95 

88 

68 

60 

24 

37 

79 

52 

52 

33 

25 

12 

98 

87 

71 

59 

23 

38 

82 

52 

55 

33 

28 

11 

1501 

87 

73 

59 

22 

39 

85 

51 

58 

33 

31 

11 

04 

86 

76 

59 

21 

40 

.0987 

.9951 

.1161 

.9932' 

.1334 

.9911 

.1507 

.9886 

.1679 

.9858 

20 

41 

90 

51 

64 

32 

37 

10 

10 

85 

82 

58 

19 

42 

93 

51 

67 

32 

40 

10 

13 

85 

85 

57 

18 

43 

96 

50 

70 

31 

43 

09 

15 

84 

88 

57 

17 

44 

99 

50 

72 

31 

46 

09 

18 

84 

91 

56 

16 

45 

.1002 

.9950 

.1175 

.9931 

.1349 

.9909 

.1521 

.9884 

.1693 

.9856 

15 

46 

05 

49 

78 

30 

51 

08 

24 

83 

96 

55 

14 

47 

08 

49 

81 

30 

54 

08 

27 

83 

99 

55 

13 

48 

11 

49 

84 

30 

57 

07 

30 

82 

1702 

54 

12 

49 

13 

49 

87 

29 

60 

07 

33 

82 

05 

54 

11 

50 

.1016 

.9948 

.1190 

.9929 

.1363 

.9907 

.1536 

.9881 

.1708 

.9853 

10 

51 

19 

48 

93 

29 

66 

06 

38 

81 

11 

53 

9 

52 

22 

48 

96 

28 

69 

06 

41 

80 

14 

52 

8 

53 

25 

47 

98 

28 

72 

05 

44 

80 

16 

52 

7 

54 

28 

47 

1201 

28 

74 

05 

47 

80 

19 

51 

6 

55 

.1031 

.9947 

.1204 

.9927 

.1377 

.9905 

.1550 

.9879 

.1722 

.9851 

5 

56 

34 

46 

07 

27 

80 

04 

53 

79 

25 

50 

4 

57 

37 

46 

10 

27 

83 

04 

56 

78 

28 

50 

3 

58 

39 

46 

13 

26 

86 

03 

59 

78 

31 

49 

2 

59 

42 

46 

16 

26 

89 

03 

61 

77 

34 

49 

1 

60 

.1045 

.9945 

.1219 

.9925 

.1392 

.9903 

.1.564 

.9877 

.1736 

.9848 

0 

COS 

sin 

COS 

sin 

cos 

sin 

COS 

sin 

COS 

sin 

r 

84 

" 

83 

o 

82 

o 

81 

o 

80 

" 

; 

78 


Natural  Tangents  and  Cotangents 


/ 

5^ 

6° 

7° 

8" 

9° 

/ 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

0 

.0875 

11.4301 

.1051 

9.5144 

.1228 

8.1443 

.1405 

7.1154 

.1584 

6.3138 

60 

1 

78 

3919 

54 

4878 

31 

1248 

08 

1004 

87 

3019 

59 

2 

81 

3540 

57 

4614 

34 

1054 

11 

0855 

90 

2901 

58 

3 

84 

3163 

60 

4352 

37 

0860 

14 

0706 

93 

2783 

57 

4 

87 

2789 

63 

4090 

40 

0667 

17 

0558 

96 

2666 

56 

5 

.0890 

11.2417 

.1066 

9.3831 

.1243 

8.0476 

.1420 

7.0410 

.1599 

6.2549 

55 

6 

92 

2048 

69 

3572 

46 

0285 

23 

0264 

1602 

2432 

54 

7 

95 

1681 

72 

3315 

49 

0095 

26 

0117 

05 

2316 

53 

8 

98 

1316 

75 

3060 

51 

7.9906 

29 

6.9972 

08 

2200 

52 

9 

0901 

0954 

78 

2806 

54 

9718 

32 

9827 

11 

2085 

51 

10 

.0904 

11.0594 

.1080 

9.2553 

.1257 

7.9530 

.1435 

6.9682 

.1614 

6.1970 

50 

11 

07 

0237 

83 

2302 

60 

9344 

38 

9538 

17 

1856 

49 

12 

10 

10.9882 

86 

2052 

63 

9158 

41 

9395 

20 

1742 

48 

13 

13 

9529 

89 

1803 

66 

8973 

44 

9252 

23 

1628 

47 

14 

16 

9178 

92 

1555 

69 

8789 

47 

9110 

26 

1515 

46 

15 

.0919 

10.8829 

.1095 

9.1309 

.1272 

7.8606 

.1450 

6.8969 

.1629 

6.1402 

45 

16 

22 

8483 

98 

1065 

75 

8424 

53 

8828 

32 

1290 

44 

17 

25 

8139 

1101 

0821 

78 

8243 

56 

8687 

35 

1178 

43 

18 

28 

7797 

04 

0579 

81 

8062 

59 

8548 

38 

1066 

42 

19 

31 

7457 

07 

0338 

84 

7882 

62 

8408 

41 

0955 

41 

20 

.0934 

10.7119 

.1110 

9.0098 

.1287 

7.7704 

.1465 

6.8269 

.1644 

6.0844 

40 

21 

36 

6783 

13 

8.9860 

90 

7525 

68 

8131 

47 

0734 

39 

22 

39 

6450 

16 

9623 

93 

7348 

71 

7994 

50 

0624 

38 

23 

42 

6118 

19 

9387 

96 

7171 

74 

7856 

53 

0514 

37 

24 

45 

5789 

22 

9152 

99 

6996 

77 

7720 

55 

0405 

36 

25 

.0948 

10.5462 

.1125 

8.8919 

.1302 

7.6821 

.1480 

6.7584 

.1658 

6.0296 

35 

26 

51 

5136 

28 

8686 

05 

6647 

83 

7448 

61 

0188 

34 

27 

54 

4813 

31 

8455 

OS 

6473 

86 

7313 

64 

0080 

33 

28 

57 

4491 

33 

8225 

11 

6301 

89 

7179 

67 

5.9972 

32 

29 

60 

4172 

36 

7996 

14 

6129 

92 

704i 

70 

9865 

31 

30 

.0963 

10.3854 

.1139 

8.7769 

.1317 

7.5958 

.1495 

6.6912 

.1673 

5.9758 

30 

31 

66 

3538 

42 

7542 

19 

5787 

97 

6779 

76 

9651 

29 

32 

69 

3224 

45 

7317 

22 

5618 

1500 

6646 

79 

9545 

28 

33 

72 

2913 

48 

7093 

25 

5449 

03 

6514 

82 

9439 

27 

34 

75 

2602 

51 

6870 

28 

5281 

06 

6383 

85 

9333 

26 

35 

.0978 

10.2294 

.1154 

8.6648 

.1331 

7.5113 

.1509 

6.6252 

.1688 

5.9228 

25 

36 

81 

1988 

57 

6427 

34 

4947 

12 

6122 

91 

9124 

24 

37 

83 

1683 

60 

6208 

37 

4781 

15 

5992 

94 

9019 

23 

38 

86 

1381 

63 

5989 

40 

4615 

18 

5863 

97 

8915 

22 

39 

89 

1080 

66 

5772 

43 

4451 

21 

5734 

1700 

8811 

21 

40 

.0992 

10.0780 

.1169 

8.5555 

.1346 

7.4287 

.1524 

6.5606 

.1703 

5.8708 

20 

41 

95 

0483 

72 

5340 

49 

4124 

27 

5478 

06 

8605 

19 

42 

98 

0187 

75 

5126 

52 

3962 

30 

5350 

09 

8502 

18 

43 

1001 

9.9893 

78 

4913 

55 

3S00 

33 

5223 

12 

8400 

17 

44 

04 

9601 

81 

4701 

58 

3639 

36 

5097 

15 

8298 

16 

45 

.1007 

9.9310 

.1184 

8.4490 

.1361 

7.3479 

.1539 

6.4971 

.1718 

5.8197 

15 

46 

10 

9021 

87 

4280 

64 

3319 

42 

4846 

21 

8095 

14 

47 

13 

8734 

89 

4071 

67 

3160 

45 

4721 

24 

7994 

13 

48 

16 

8448 

92 

3863 

70 

3002 

48 

4596 

27 

7894 

12 

49 

19 

8164 

95 

3656 

73 

2844 

51 

4472 

30 

7794 

11 

50 

.1022 

9.7882 

.1198 

8.3450 

.1376 

7.2687 

.1554 

6.4348 

.1733 

5.7694 

10 

51 

25 

7601 

1201 

3245 

79 

2.531 

57 

4225 

36 

7594 

9 

52 

28 

7322 

04 

3041 

82 

2375 

60 

4103 

39 

7495 

8 

53 

30 

7044 

07 

2838 

85 

2220 

63 

3980 

42 

7396 

7 

54 

33 

6768 

10 

2636 

88 

2066 

66 

3859 

45 

7297 

6 

55 

.1036 

9.6493 

.1213 

8.2434 

.1391 

7.1912 

.1569 

6.3737 

.1748 

5.7199 

5 

56 

39 

6220 

16 

2234 

94 

1759 

72 

3617 

51 

7101 

4 

57 

42 

5949 

19 

2035 

97 

1607 

75 

3496 

54 

7004 

3 

58 

45 

5679 

22 

1837 

99 

1455 

78 

3376 

57 

6906 

2 

59 

48 

5411 

25 

1640 

1402 

1304 

81 

3257 

60 

6809 

1 

60 

.1051 

9.5144 

.1228 

8.1443 

.1405 

7.1154 

.1584 

6.3138 

.1763 

5.6713 

0 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

/ 

84° 

83° 

82° 

81° 

80° 

t 

79 


Natural  Sines  and  Cosines 


1 

10 

^ 

11 

=^ 

12 

o 

13°#   1 

14' 

/ 

sin 

cos 

sin 

cos 

sin 

COS 

sin   COS 

sin 

COS 

0 

.1736 

.9848 

.1908 

.9816 

.2079 

.9781 

.2250  .9744 

.2419 

9703 

60 

1 

39 

48 

11 

16 

82 

81 

52    43 

22 

02 

59 

2 

42 

47 

14 

15 

8i 

80 

55    42 

25 

02 

58 

3 

45 

47 

17 

15 

88 

80 

58    42 

28 

01 

57 

4 

48 

46 

20 

14 

90 

79 

61    41 

31 

00 

56 

6 

.1751 

.9846 

.1922 

.9813 

.2093 

.9778 

.2264  .9740 

.2433 

9699 

55 

6 

54 

45 

25 

13 

96 

78 

67    40 

36 

99 

54 

7 

57 

4i 

28 

12 

99 

77 

69    39 

39 

98 

53 

8 

59 

44 

31 

12 

2102 

77 

72    38 

42 

97 

52 

9 

62 

43 

34 

11 

05 

76 

75    38 

45 

97 

51 

10 

.1765 

.9843 

.1937 

.9811 

.2108 

.9775 

.2278  .9737 

.2447 

.9696 

50 

11 

68 

42 

39 

10 

10 

75 

81    36 

50 

95 

49 

12 

71 

42 

42 

10 

13 

74 

84    36 

53 

94 

48 

13 

74 

41 

45 

09 

16 

74 

86    35 

56 

94 

47 

14 

77 

41 

48 

08 

19 

73 

89    34 

59 

93 

46 

15 

.1779 

.9840 

.1951 

.9808 

.2122 

.9772 

.2292  .9734 

.2462 

.9692 

45 

16 

82 

40 

54 

07 

25 

72- 

95    33 

64 

92 

44 

17 

85 

39 

57 

07 

27 

71 

98    32 

67 

91 

43 

18 

88 

39 

59 

06 

30 

70 

2300    32 

70 

90 

42 

19 

91 

38 

62 

06 

33 

70 

03    31 

73 

89 

41 

20 

.1794 

.9838 

.1965 

.9805 

.2136 

.9769 

.2306  .9730 

.2476 

.9689 

40 

21 

97 

37 

68 

04 

39 

69 

09    30 

78 

88 

39 

22 

99 

37 

71 

04 

42 

68 

12    29 

81 

87 

38 

23 

1802 

36 

74 

03 

45 

67 

15    28 

84 

87 

37 

24 

05 

36 

77 

03 

47 

67 

17    28 

87 

86 

36 

25 

.1808 

.9835 

.1979 

.9802 

.2150 

.9766 

.2320  .9727- 

.2490 

.9685 

35 

26 

11 

35 

82 

02 

53 

65 

23    26 

93 

84 

34 

27 

14 

34 

85 

01 

56 

65 

26    26 

95 

84 

33 

28 

17 

34 

88 

00 

59 

64 

29    25 

98 

83 

32 

29 

19 

33 

91 

00 

62 

64 

32    24 

.2501 

82 

31 

30 

.1822 

.9833 

.1994 

.9799 

.2164 

.9763 

.2334  .9724 

.2504 

.%81 

30 

31 

25 

32 

97 

99 

67 

62 

37    23 

07 

81 

29 

32 

28 

31 

99 

98 

70 

62 

40    22 

09 

80 

28 

33 

31 

31 

2002 

98 

73 

61 

43    22 

12 

79 

27 

34 

34 

30 

05 

97 

76 

60 

46    21 

15 

79 

26 

35 

.1837 

.9830 

.2008 

.9796  . 

.2179 

.9760 

.2349  .9720 

.2518 

.9678 

25 

36 

40 

29 

11 

% 

81 

59 

51    20 

21 

77 

24 

37 

42 

29 

14 

95 

84 

59 

54    19 

24 

76 

23 

38 

45 

28 

16 

95 

87 

58 

57    18 

26 

76 

22 

39 

48 

28 

19 

94 

90 

57 

60    18 

29 

75 

21 

40 

.1851 

.9827 

.2022 

.9793 

.2193 

.9757 

.2363  .9717 

.2532 

.9674 

20 

41 

54 

27 

25 

93 

% 

56 

66    16 

35 

73 

19 

42 

57 

26 

28 

92 

98 

55 

68    15 

38 

73 

18 

43 

60 

26 

31 

92 

2201 

55 

71    15 

40 

72 

17 

44 

62 

25 

34 

91 

04 

54 

74    14 

43 

71 

16 

45 

.1865 

.9825 

.2036 

.9790 

.2207 

.9753 

.2377  .9713 

.2546 

.9670 

15 

46 

68 

24 

39 

90 

10 

53 

80    13 

49 

70 

14 

47 

71 

23 

42 

89 

13 

52 

83    12 

52 

69 

13 

48 

74 

23 

45 

89 

15 

51 

85    11 

54 

68 

12 

49 

77 

22 

48 

88 

18 

51 

88    11 

57 

67 

11 

50 

.1880 

.9822 

.2051 

.9787 

.2221 

.9750 

.2391  .9710 

.2560 

.9667 

10 

51 

82 

21 

54 

87 

24 

50 

94    09 

63 

66 

9 

52 

85 

21 

56 

86 

27 

49 

97    09 

66 

65 

8 

53 

88 

20 

59 

86 

30 

48 

99    08 

69 

65 

7 

54 

91 

20 

62 

85 

33 

48 

.2402    07 

71 

64 

6 

65 

.1894 

.9819 

.2065 

.9784 

.2235 

.9747 

.2405  .9706 

.2574 

.9663 

5 

56 

97 

18 

68 

84 

38 

46 

08    06 

77 

62 

4 

57 

1900 

18 

71 

83 

41 

46 

11    05 

80 

62 

3 

58 

02 

17 

73 

83 

44 

45 

14    04 

83 

61 

2 

59 

05 

17 

76 

82 

47 

44 

16   04 

85 

60 

1 

60 

.1908 

.9816 

.2079 

.9781 

.2250 

.9744 

.2419  .9703 

.2588 

.9659 

0 

COS 

sin 

COS 

sin 

cos 

sin 

COS   sin 

COS 

sin 

/ 

/ 

79" 

78° 

77° 

76° 

75^ 

80 


Katural  Tangents  and  Cotangents 


1 

10° 

11° 

12° 

13° 

14° 

1 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

0 

.1763  5.6713 

.1944  5.1446 

.2126  4.7046 

.2309  4.3315 

.2493  4.0108 

60 

1 

66   6617 

47   1366 

29   6979 

12   3257 

96   0058 

59 

2 

69   6521 

50   1286' 

32   6912 

15   3200 

99   0009 

58 

3 

72   6425 

53   1207 

35   6845 

18   3143 

2503  3.9959 

57 

4 

75   6329 

56   1128 

38   6779 

21   3086 

06   9910 

56 

5 

.1778  5.6234 

.1959  5.1049 

.2141  4.6712 

.2324  4.3029 

.2509  3.9861 

55 

6 

81   6140 

62   0970 

44   6646 

27   2972 

12   9812 

54 

7 

84   6045 

65   0892 

47   6580 

30   2916 

15   9763 

53 

S 

87   5951 

68   0814 

50   6514 

33   2859 

18   9714 

52 

9 

90   5857 

71   0736 

53   6448 

36   2803 

21   9665 

51 

10 

.1793  5.5764 

.1974  5.0658 

.21.56  4.6382 

.2339  4.2747 

.2524  3.9617 

50 

11 

96   5671 

77   0581 

59   6317 

42   2691 

27   9568 

49 

12 

99   5578 

80  0504 

62   6252 

45   2635 

30   9520 

48 

13 

1802   5485 

83   0427 

65   6187 

49   2580 

33   9471 

47 

14 

05   5393 

86   0350 

68   6122 

52   2524 

37   9423 

46 

15 

.1808  5.5301 

.1989  5.0273 

.2171  4.6057 

.2355  4.2468 

.2540  3.9375 

45 

16 

11   5209 

92   0197 

74   5993 

58   2413 

43   9327 

44 

17 

14   5118 

95   0121 

77   5928 

61   2358 

46   9279 

43 

18 

17   5026 

98   0045 

80   5864 

64   2303 

49   9232 

42 

19 

20   4936 

2001  4.9969 

83   5800 

67   2248 

52   9184 

41 

20 

.1823  5.4845 

.2004  4.9894 

.2186  4.5736 

.2370  4.2193 

.2555  3.9136 

40 

21 

26   4755 

07   9819 

89   5673 

73   2139 

58   9089 

39 

22 

29   4665 

10   9744 

93   5609 

76   2084 

61   9042 

38 

23 

32   4575 

13   9669 

96   5546 

79   2030 

64   8995 

37 

24 

35   4486 

16   9594 

99   5483 

82   1976 

68   8947 

36 

25 

.1838  5.4397 

.2019  4.9520 

.2202  4.5420 

.2385  4.1922 

.2571  3.8900 

35 

26 

41   4308 

22   9446 

05   5357 

88   1868 

74   8854 

34 

27 

44   4219 

25   9372 

08   5294 

92   1814 

77   8807 

33 

28 

47   4131 

28   9298 

11   5232 

95   1760 

80   8760 

32 

29 

50   4043 

31   9225 

14   5169 

98   1706 

83   8714 

31 

30 

.1853  5.3955 

.2035  4.9152 

.2217  4.5107 

.2401  4.1653 

.2586  3.8667 

30 

31 

56   3868 

38   9078 

20   5045 

04   1600 

89   8621 

29 

32 

59   3781 

41   9006 

23   4983 

07   1.547 

92   8575 

28 

33 

62   3694 

44   8933 

26   4922 

10   1493 

95   8528 

27 

34 

65   3607 

47   8860 

29   4860 

13   1441 

99   8482 

26 

35 

.1868  5.3521 

.2050  4.8788 

.2232  4.4799 

.2416  4.1388 

.2602  3.8436 

25 

36 

71   3435 

53   8716 

35   4737 

19   1335 

05   8391 

24 

37 

74   3349 

56   8644 

38   4676 

22   1282 

08   8345 

23 

38 

77   3263 

59   8573 

41   4615 

25   1230 

11   8299 

22 

39 

80  3178 

62   8501 

44   4555 

28   1178 

14   8254 

21 

40 

.1883  5.3093 

.2065  4.8430 

.2247  4.4494 

.2432  4.1126 

.2617  3.8208 

20 

41 

87   3008 

68   8359 

51   4434 

35   1074 

20   8163 

19 

42 

90   2924 

71   8288 

54   4373 

38   1022 

23   8118 

18 

43 

93   2839 

74  8218 

57   4313 

41   0970 

27   8073 

17 

44 

96   2755 

77  8147 

60   4253 

44  0918 

30   8028 

16 

45 

.1899  5.2672 

.2080  4.8077 

.2263  4.4194 

.2447  4.0867 

.2633  3.7983 

15 

46 

1902   2588 

83   8007 

66   4134 

50  0815 

36   7938 

14 

47 

05   2505 

86  7937 

69   4075 

53   0764 

39   7893 

13 

48 

08   2422 

89  7867 

72   4015 

56  0713 

42   7848 

12 

49 

11   2339 

92   7798 

75   3956 

59   0662 

45   7804 

11 

50 

.1914  5.2257 

.2095  4.7729 

.2278  4.3897 

.2462  4.0611 

.2648  3.7760 

10 

51 

17   2174 

98   7659 

81   3838 

65   0560 

51   7715 

9 

52 

20   2092 

2101   7591 

84  3779 

69   0509 

55   7671 

8 

53 

23   2011 

04   7522 

87   3721 

72   0459 

58   7627 

7 

54 

26   1929 

07   7453 

90  3662 

75   0408 

61   7583 

6 

55 

.1929  5.1848 

.2110  4.7385 

.2293  4.3604 

.2478  4.0358 

.2664  3.7539 

5 

56 

32   1767 

13   7317 

96   3546 

81   0308 

67   7495 

4 

57 

35   1686 

16   7249 

99   3488 

84  0257 

70   7451 

3 

58 

38   1606 

19   7181 

2303   3430 

87   0207 

73   7408 

2 

59 

41   1526 

23   7114 

06   3372 

90   0158 

76   7364 

1 

60 

.1944  5.1446 

.2126  4.7046 

.2309  4.3315 

.2493  4.0108 

.2679  3.7321 

0 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

/ 

79° 

78° 

77° 

76° 

75° 

1 

81 


Natural  Sines  and  Cosines 


/ 

15" 

16= 

17° 

18" 

19° 

( 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

0 

.2588 

.%59 

.2756 

.9613 

.2924 

.9563 

.3090 

.9511 

.3256 

.9455 

60 

1 

91 

59 

59 

12 

26 

62 

93 

10 

58 

54 

59 

2 

94 

58 

62 

11 

29 

61 

96 

09 

61 

53 

58 

3 

97 

57 

65 

10 

32 

60 

98 

08 

64 

52 

57 

4 

99 

56 

68 

09 

3i 

60 

3101 

07 

67 

51 

56 

5 

.2602 

.9655 

.2770 

.9609 

.2938 

.9559 

.3104 

.9506 

.3269 

.9450 

55 

6 

05 

55 

73 

08 

40 

58 

07 

05 

72 

49 

54 

7 

08 

54 

76 

07 

43 

57 

10 

04 

75 

49 

53 

8 

11 

•53 

79 

06 

46 

56 

12 

03 

78 

48 

52 

9 

13 

52 

82 

05 

49 

55 

15 

02 

80 

47 

51 

10 

.2616 

.9652 

.2784 

.9605 

.2952 

.955i 

.3118 

.9502 

.3283 

.9446 

50 

11 

19 

51 

87 

04 

54 

54 

21 

01 

86 

45 

49 

12 

22 

50 

90 

03 

57 

53 

23 

9i00 

89 

44 

48 

13 

2i 

49 

93 

02 

60 

52 

26 

9499 

91 

43 

47 

14 

28 

49 

95 

01 

63 

51 

29 

98 

94 

42 

46 

15 

.2630 

.9648 

.2798 

.9600 

.2965 

.9550 

.3132 

.9497 

.3297 

.9441 

45 

16 

33 

47 

2801 

00. 

68 

49 

34 

96 

3300 

40 

44 

17 

36 

46 

04 

9599 

71 

48 

37 

95 

02 

39 

43 

18 

39 

46 

07 

98 

74 

48 

40 

94 

05 

38 

42 

19 

^2 

4i 

09 

97 

77 

^7 

43 

93 

08 

37 

41 

20 

.2644 

.9644 

.2812 

.9596 

.2979 

.9546 

.3145 

.9492 

.3311 

.9436 

40 

21 

47 

43 

15 

96 

82 

45 

48 

92 

13 

35 

39 

22 

50 

42 

18 

95 

85 

44 

51 

91 

16 

34 

38 

23 

53 

42 

21 

94 

88 

43 

54 

90 

19 

33 

37 

24 

56 

41 

23 

93 

90 

42 

56 

89 

22 

32 

36 

26 

.2658 

.9640 

.2826 

.9592 

.2993 

.9542 

.3159 

.9488 

.3324 

.9431 

35 

26 

61 

39 

29 

91 

96 

41 

62 

87 

27 

30 

34 

27 

64 

39 

32 

91 

99 

40 

65 

86 

30 

29 

33 

28 

67 

38 

35 

90 

3002 

39 

68 

85 

33 

28 

32 

29 

70 

37 

37 

89 

04 

38 

70 

84 

35 

27 

31 

30 

.2672 

.9636 

.2840 

.9588 

.3007 

.9537 

.3173 

.9483 

.3338 

.9426 

30 

31 

75 

36 

43 

87 

10 

36 

76 

82 

41 

25 

29 

32 

78 

35 

46 

87 

13 

35 

79 

81 

44 

24 

28 

33 

81 

34 

49 

86 

15 

35 

81 

80 

46 

23 

27 

34 

84 

33 

51 

85 

18 

34 

84 

80 

49 

23 

26 

35 

.2686 

.9632 

.2854 

.9584 

.3021 

.9533 

.3187 

.9479 

.3352 

.9422 

25 

36 

89 

32 

57 

83 

24 

32 

90 

78 

55 

21 

24 

37 

92 

31 

60 

82 

26 

31 

92 

77 

57 

20 

23 

38 

95 

30 

62 

82 

29 

30 

95 

76 

60 

19 

22 

39 

98 

29 

65 

81 

32 

29 

98 

7i 

63 

18 

21 

40 

.2700 

.9628 

.2868 

.9580 

.3035 

.9528 

.3201 

.9474 

.3365 

.9417 

20 

41 

03 

28 

71 

79 

38 

27 

03 

73 

68 

16 

19 

42 

06 

27 

74 

78 

40 

27 

06 

72 

71 

15 

18 

43 

09 

26 

76 

77 

43 

26 

09 

71 

74 

14 

17 

44 

12 

25 

79 

77 

46 

25 

12 

70 

76 

13 

16 

45 

.2714 

.9625 

.2882 

.9576 

.3049 

.9524 

.3214 

.9469 

.3379 

.9412 

15 

46 

17 

24 

85 

75 

51 

23 

17 

68 

82 

11 

14 

47 

20 

23 

88 

74 

54 

22 

20 

67 

85 

10 

13 

48 

23 

22 

90 

73 

57 

21 

23 

66 

87 

09 

12 

49 

26 

21 

93 

72 

60 

20 

25 

66 

90 

08 

11 

50 

2728 

.9621 

.2896 

.9572 

.3062 

.9520 

.3228 

.9465 

.3393 

.9407 

10 

51 

31 

20 

99 

71 

65 

19 

31 

64 

96 

06 

9 

52 

34 

19 

2901 

70 

68 

18 

34 

63 

98 

05 

8 

53 

37 

18 

04 

69 

71 

17 

36 

62 

3401 

04 

7 

54 

40 

17 

07 

68 

74 

16 

39 

61 

04 

03 

6 

55 

.2742 

.9617 

.2910 

.9567 

.3076 

.9515 

.3242 

.9460 

.3407 

.9402 

5 

56 

45 

16 

13 

66 

79 

14 

45 

59 

09 

01 

4 

57 

48 

15 

15 

66 

82 

13 

47 

58 

12 

00 

3 

58 

51 

14 

18 

65 

85 

12 

50 

57 

li 

9399 

2 

59 

54 

13 

21 

64 

87 

11 

53 

56 

17 

98 

1 

60 

.2756 

.%13 

.2924 

.9563 

.3090 

.9511 

.3256 

.9455 

.3420 

.9397 

0 

COS 

sin 

COS 

sin 

cos 

sin 

COS 

sin 

cos 

sin 

/ 

74 

o 

73 

° 

72 

o 

71 

0 

70 

" 

t 

82 


Natural  Tangents  and  Cotangents 


/ 

15° 

16° 

17° 

18° 

19°     1 

1 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

0 

.2679 

3.7321 

.2867 

3.4874 

.3057 

3.2709 

.3249 

3.0777 

.3443 

2.9042 

60 

1 

83 

7277 

71 

4836 

60 

2675 

52 

0746 

47 

9015 

59 

2 

86 

7234 

74 

4798 

64 

2641 

56 

0716 

50 

8987 

58 

3 

89 

7191 

77 

4760 

67 

2607 

59 

0686 

53 

8960 

57 

4 

92 

7148 

80 

4722 

70 

2573 

62 

0655 

56 

8933 

56 

5 

.2695 

3.7105 

.2883 

3.4684 

.3073 

3.2539 

.3265 

3.0625 

.3460 

2.8905 

55 

6 

98 

7062 

86 

4646 

76 

2506 

69 

0595 

63 

8878 

54 

7 

2701 

7019 

90 

4608 

80 

2472 

72 

0565 

66 

8851 

53 

8 

(H 

6976 

93 

4570 

83 

2438 

75 

0535 

69 

8824 

52 

9 

08 

6933 

96 

4533 

86 

2405 

78 

0505 

73 

8797 

51 

10 

.2711 

3.6891 

.2899 

3.4495 

.3089 

3.2371 

.3281 

3.0475 

.3476 

2.8770 

50 

11 

14 

6848 

2902 

4458 

92 

2338 

85 

0145 

79 

8743 

49 

12 

17 

6806 

05 

4420 

96 

2305 

88 

0415 

82 

8716 

48 

13 

20 

6764 

08 

4383 

99 

2272 

91 

0385 

86 

8689 

47 

14 

23 

6722 

12 

4346 

3102 

2238 

94 

0356 

89 

8662 

46 

15 

.2726 

3.6680 

.2915 

3.4308 

.3105 

3.2205 

.3298 

3.0326 

.3492 

2.8636 

45 

16 

29 

6638 

18 

4271 

08 

2172 

3301 

0296 

95 

8609 

44 

17 

33 

65% 

21 

4234 

11 

2139 

04 

0267 

99 

8582 

43 

IS 

36 

6554 

24 

4197 

15 

2106 

07 

0237 

3502 

8556 

42 

19 

39 

6512 

27 

4160 

18 

2073 

10 

0208 

05 

8529 

41 

20 

.2742 

3.6470 

.2931 

3.4124 

.3121 

3.2041 

.3314 

3.0178 

.3508 

2.8502 

40 

21 

45 

6429 

34 

4087 

24 

2008 

17 

0149 

12 

8476 

39 

22 

48 

6387 

37 

4050 

27 

1975 

20 

0120 

15 

8449 

38 

23 

51 

6346 

40 

4014 

31 

1943 

23 

0090 

18 

8423 

37 

24 

54 

6305 

43 

3977 

34 

1910 

27 

0061 

22 

8397 

36 

25 

.2758 

3.6264 

.2946 

3.3941 

.3137 

3.1878 

.3330 

3.0032 

.3525 

2.8370 

35 

26 

61 

6222 

49 

3904 

40 

1845 

33 

0003 

28 

8344 

34 

27 

64 

6181 

53 

3868 

43 

1813 

36 

2.9974 

31 

8318 

33 

28 

67 

6140 

56 

3832 

47 

1780 

39 

9945 

35 

8291 

32 

29 

70 

6100 

59 

3796 

50 

1748 

43 

9916 

38 

8265 

31 

30 

.2773 

3.6059 

.2962 

3.3759 

.3153 

3.1716 

.3346 

2.9887 

.3541 

2.8239 

30 

31 

76 

6018 

65 

3723 

56 

1684 

49 

9858 

44 

8213 

29 

32 

80 

5978 

68 

3687 

59 

1652 

52 

9829 

48 

8187 

28 

33 

83 

5937 

72 

3652 

63 

1620 

56 

9800 

51 

8161 

27 

34 

86 

5897 

75 

3616 

66 

1588 

59 

9772 

54 

8135 

26 

35 

.2789 

3.5856 

.2978 

3.3580 

.3169 

3.1556 

.3362 

2.9743 

.3558 

2.8109 

25 

36 

92 

5816 

81 

3544 

72 

1524 

65 

9714 

61 

8083 

24 

37 

95 

5776 

84 

3509 

75 

1492 

69 

9686 

64 

8057 

23 

38 

98 

5736 

87 

3473 

79 

1460 

72 

9657 

67 

8032 

22 

39 

2801 

5696 

91 

3438 

82 

1429 

75 

9629 

71 

8006 

21 

40 

.2805 

3.5656 

.2994 

3.3402 

.3185 

3.1397 

.3378 

2.9600 

.3574 

2.7980 

20 

41 

08 

5616 

97 

3367 

88 

1366 

82 

9572 

77 

7955 

19 

42 

11 

5576 

3000 

3332 

91 

1334 

85 

9544 

81 

7929 

18 

43 

14 

5536 

03 

3297 

95 

1303 

88 

9515 

84 

7903 

17 

44 

17 

5497 

06 

3261 

98 

1271 

91 

9487 

87 

7878 

16 

45 

.2820 

3.5457 

.3010 

3.3226 

.3201 

3.1240 

.3395 

2.9459 

.3590 

2.7852 

15 

46 

23 

5418 

13 

3191 

04 

1209 

98 

9431 

94 

7827 

14 

47 

27 

5379 

16 

3156 

07 

1178 

3401 

9403 

97 

7801 

13 

48 

30 

5339 

19 

3122 

11 

1146 

04 

9375 

3600 

7776 

12 

49 

33 

5300 

22 

3087 

14 

1115 

08 

9347 

04 

7751 

11 

50 

.2836 

3.5261 

.3026 

3.3052 

.3217 

3.1084 

.3411 

2.9319 

.3607 

2.7725 

10 

51 

39 

5222 

29 

3017 

20 

1053 

14 

9291 

10 

7700 

9 

52 

42 

5183 

32 

2983 

23 

1022 

17 

9263 

13 

7675 

8 

53 

45 

5144 

35 

2948 

27 

0991 

21 

9235 

17 

7650 

7 

54 

49 

5105 

38 

2914 

30 

0%1 

24 

9208 

20 

7625 

6 

55 

.2852 

3.5067 

.3041 

3.2879 

.3233 

3.0930 

.3427 

2.9180 

.3623 

2.7600 

5 

56 

55 

5028 

45 

2845 

36 

0899 

30 

9152 

27 

7575 

4 

57 

58 

4989 

48 

2811 

40 

0868 

34 

9125 

30 

7550 

3 

58 

61 

4951 

51 

2777 

43 

0838 

37 

9097 

33 

7525 

2 

59 

64 

4912 

54 

2743 

46 

0807 

40 

9070 

36 

7500 

1 

60 

.2867 

3.4874 

.3057 

3.2709 

.3249 

3.0777 

.3443 

2.9042 

.3640 

2.7475 

0 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

' 

74° 

73° 

72° 

71° 

70° 

/ 

83 


Natural  Sines  and  Cosines 


/ 

20" 

21 

0 

22 

c 

23 

° 

24 

0 

"1 

sin 

cos 

sin 

COS 

sin 

COS 

sin 

COS 

sin 

COS 

0 

.3420 

.9397 

.3584 

.9336 

.3746 

.9272 

.3907 

.9205 

.4067 

.9135 

60 

1 

23 

96 

86 

35 

49 

71 

10 

04 

70 

34 

59 

2 

26 

95 

89 

34 

51 

70 

13 

03 

73 

33 

58 

3 

28 

94 

92 

33 

54 

69 

15 

02 

75 

32 

57 

4 

31 

93 

95 

32 

57 

67 

18 

00 

78 

31 

56 

6 

.3434 

.9392 

.3597 

.9331 

.3760 

.9266 

.3921 

.9199 

.4081 

.9130 

55 

6 

37 

91 

3600 

30 

62 

65 

23 

98 

83 

28 

54 

7 

39 

90 

03 

28 

65 

64 

26 

97 

86 

27 

53 

8 

42 

89 

05 

27 

68 

63 

29 

96 

89 

26 

52 

9 

45 

88 

08 

26 

70 

62 

31 

95 

91 

25 

51 

10 

.3448 

.9387 

.3611 

.9325 

.3773 

.9261 

.3934 

.9194 

.4094 

.9124 

50 

11 

50 

86 

14 

24 

76 

60 

37 

92 

97 

22 

49 

12 

53 

85 

16 

23 

78 

59 

39 

91 

99 

21 

48 

13 

56 

84 

19 

22 

81 

58 

42 

90 

4102 

20 

47 

14 

58 

83 

22 

21 

84 

57 

45 

89 

05 

19 

46 

15 

.3461 

.9382 

.3624 

.9320 

.3786 

.9255 

.3947 

.9188 

.4107 

.9118 

45 

16 

64 

81 

27 

19 

89 

54 

50 

87 

10 

16 

44 

17 

67 

80 

30 

18 

92 

53 

53 

86 

12 

15 

43 

18 

69 

79 

33 

17 

95 

52 

55 

84 

15 

14 

42 

19 

72 

78 

35 

16 

97 

51 

58 

83 

18 

13 

41 

20 

.3475 

.9377 

.3638 

.9315 

.3800 

.9250 

.3961 

.-9182 

.4120 

.9112 

40 

21 

78 

76 

41 

14 

03 

49 

63 

81 

23 

10 

39 

22 

80 

75 

43 

13 

05 

48 

66 

80 

26 

09 

38 

23 

83 

74 

46 

12 

08 

47 

69 

79 

28 

08 

37 

24 

86 

73 

49 

11 

11 

45 

71 

78 

31 

07 

36 

25 

.3488 

.9372 

.3651 

.9309 

.3813 

.9244 

.3974 

.9176 

.4134 

.9106 

35 

26 

91 

71 

54 

08 

16 

43 

77 

75 

36 

04 

34 

27 

94 

70 

57 

07 

19 

42 

79 

74 

39 

03 

33 

28 

97 

69 

60 

06 

21 

41 

82 

73 

42 

02 

32 

29 

99 

68 

62 

05 

24 

40 

85 

72 

44 

01 

31 

30 

.3502 

.9367 

.3665 

.9304 

.3827 

.9239 

.3987 

.9171 

.4147 

.9100 

30 

31 

05 

66 

68 

03 

30 

38 

90 

69 

50 

9098 

29 

32 

08 

65 

70 

02 

32 

37 

93 

68 

52 

97 

28 

33 

10 

64 

73 

01 

35 

35 

95 

67 

55 

96 

27 

34 

13 

63 

76 

00 

38 

34 

98 

66 

58 

95 

26 

35 

.3516 

.9362 

.3679 

.9299 

.3840 

.9233 

.4001 

.9165 

.4160 

.9094 

25 

36 

18 

61 

81 

98 

43 

32 

03 

64 

63 

92 

24 

37 

21 

60 

84 

97 

46 

31 

06 

62 

65 

91 

23 

38 

24 

59 

87 

96 

48 

30 

09 

61 

68 

90 

22 

39 

27 

58 

89 

95 

51 

29 

11 

60 

71 

89 

21 

40 

.3529 

.9356 

.3692 

.9293 

.3854 

.9228 

.4014 

.9159 

.4173 

.9088 

20 

41 

32 

55 

95 

92 

56 

27 

17 

58 

76 

86 

19 

42 

35 

54 

97 

91 

59 

25 

19 

57 

79 

85 

18 

43 

37 

53 

3700 

90 

62. 

24 

22 

55 

81 

84 

17 

44 

40 

52 

03 

89 

64 

23 

25 

54 

84 

83 

16 

45 

.3543 

.9351 

.3706 

.9288 

.3867 

.9222 

.4027 

.9153 

.4187 

.9081 

15 

46 

46 

50 

08 

87 

70 

21 

30 

52 

89 

80 

14 

47 

48 

49 

11 

86 

72 

20 

33 

51 

92 

79 

13 

48 

51 

48 

14 

85 

75 

19 

35 

50 

95 

78 

12 

49 

54 

47 

16 

84 

78 

18 

38 

48 

97 

77 

11 

50 

.3557 

.9346 

.3719 

.9283 

.3881 

.9216 

.4041 

.9147 

.4200 

.9075 

10 

51 

59 

45 

22 

82 

83 

15 

43 

46 

02 

74 

9 

52 

62 

44 

24 

81 

86 

14 

46 

45 

05 

73 

8 

53 

65 

43 

27 

79 

89 

13 

49 

44 

08 

72 

7 

54 

67 

42 

30 

78 

91 

12 

51 

43 

10 

70 

6 

55 

.3570 

.9341 

.3733 

.9277 

.3894 

.9211 

.4054 

.9141 

.4213 

.9069 

5 

56 

73 

40 

35 

76 

97 

10 

57 

40 

16 

68 

4 

57 

76 

39 

38 

75 

99 

08 

59 

39 

18 

67 

3 

58 

78 

38 

41 

74 

3902 

07 

62 

38 

21 

66 

2 

59 

81 

37 

43 

73 

05 

06 

65 

37 

24 

64 

1 

60 

.3584 

.9336 

.3746 

.9272 

.3907 

.9205 

.4067 

.9135 

.4226 

.9063 

0 

cos 

sin 

COS 

sin 

COS 

sin 

COS 

sin 

COS 

sin 

/ 

69^ 

68° 

67° 

66° 

65' 

/ 

84 


Natural  Tangents  and  Cotangents 


1 

20° 

21° 

22° 

23° 

24° 

1 

tan   cot 

tan   cot 

tan   cot 

tan 

cot 

tan   cot 

0 

.3640  2.7475 

.3839  2.6051 

.4040  2.4751 

.4245 

2.3559 

.4452  2.2460 

60 

1 

43   7450 

42   6028 

44   4730 

48 

3539 

56   2443 

59 

2 

46   7425 

45   6006 

47   4709 

52 

3520 

59   2425 

58 

3 

50   7400 

49   5983 

50   4689 

55 

3501 

63   2408 

57 

4 

53   7376 

52   5961 

54   4668 

58 

3483 

66   2390 

56 

5 

.3656  2.7351 

.3855  2.5938 

.4057  2.4648 

.4262 

2.3464 

.4470  2.2373 

55 

6 

59   7326 

59   5916 

61   4627 

65 

3445 

73   2355 

54 

7 

63   7302 

62   5893 

64   4606 

69 

3426 

77   2338 

53 

8 

66   7277 

65   5871 

67   4586 

72 

3407 

80   2320 

52 

9 

69   7253 

69   5848 

71   4566 

76 

3388 

84   2303 

51 

10 

.3673  2.7228 

.3872  2.5826 

.4074  2.4545 

.4279  2.3369 

.4487  2.2286 

50 

11 

76   7204 

75   5804 

78   4525 

83 

3351 

91   2268 

49 

12 

79   7179 

79   5782 

81   4504 

86 

3332 

94   2251 

48 

13 

83   7155 

82   5759 

84   4484 

89 

3313 

98   2234 

47 

14 

86   7130 

85   5737 

88   4464 

93 

3294 

4501   2216 

46 

15 

.3689  2.7106 

.3889  2.5715 

.4091  2.4443 

.4296  2.3276 

.4505  2.2199 

45 

16 

93   7082 

92   5693 

95   4423 

4300 

3257 

08   2182 

44 

17 

96   7058 

95   5671 

98   4403 

03 

3238 

12   2165 

43 

18 

99   7034 

99   5649 

4101   4383 

07 

3220 

15   2148 

42 

19 

3702   7009 

3902   5627 

05   4362 

10 

3201 

19   2130 

41 

20 

.3706  2.6985 

.3906  2.5605 

.4108  2.4342 

.4314 

2.3183 

.4522  2.2113 

40 

21 

09   6961 

09   5583 

11   4322 

17 

3164 

26   2096 

39 

22 

12   6937 

12   5561 

15   4302 

20 

3146 

29   2079 

38 

23 

16   6913 

16   5539 

18   4282 

24 

3127 

33   2062 

37 

24 

19   6889 

19   5517 

22   4262 

27 

3109 

36   2045 

36 

25 

.3722  2.6865 

.3922  2.5495 

.4125  2.4242 

.4331 

2.3090 

.4540  2.2028 

35 

26 

26   6841 

26   5473 

29   4222 

34 

3072 

43   2011 

34 

27 

29   6818 

29   5452 

32   4202 

38 

3053 

47   1994 

33 

28 

32   6794 

32   5430 

35   4182 

41 

3035 

50   1977 

32 

29 

36   6770 

36   5408 

39   4162 

45 

3017 

54   1960 

31 

30 

.3739  2.6746 

.3939  2.5386 

.4142  2.4142 

.4348 

2.2998 

.4557  2.1943 

30 

31 

42   6723 

42   5365 

46   4122 

52 

2980 

61   1926 

29 

32 

45   6699 

46   5343 

49   4102 

55 

2962 

64   1909 

28 

33 

49   6675 

49   5322 

52   4€83 

59 

2944 

68   1892 

27 

34 

52   6652 

53   5300 

56   4063 

62 

2925 

71   1876 

26 

35 

.3755  2.6628 

.3956  2.5279 

.4159  2.4043 

.4365 

2.2907 

.4575  2.1859 

25 

36 

59   6605 

59  5257 

63   4023 

69 

2889 

78   1842 

24 

37 

62   6581 

63   5236 

66   4004 

72 

2871 

82   1825 

23 

38 

65   6558 

66   5214 

69   3984 

76 

2853 

85   1808 

22 

39 

69   6534 

69   5193 

73   3964 

79 

2835 

89   1792 

21 

40 

.3772  2.6511 

.3973  2.5172 

.4176  2.3945 

.4383 

2.2817 

.4592  2.1775 

20 

41 

75   6488 

76  5150 

80   3925 

86 

2799 

96   1758 

19 

42 

79   6464 

79   5129 

83   3906 

90 

2781 

99   1742 

18 

43 

82   6441 

83   5108 

87   3886.' 

'   93 

2763 

4603   1725 

17 

44 

85   6418 

86  5086 

90   3867 

97 

2745 

07   1708 

16 

45 

.3789  2.6395 

.3990  2.5065 

.4193  2.3847 

.4400  2.2727 

.4610  2.1692 

15 

46 

92   6371 

93   5044 

97   3828 

04 

2709 

14   1675 

14 

47 

95   6348 

96   5023 

4200   3808 

07 

2691 

17   1659 

13 

48 

99   6325 

4000   5002 

04   3789 

11 

2673 

21   1642 

12 

49 

3802   6302 

03   4981 

07   3770 

14 

2655 

24   1625 

11 

60 

.3805  2.6279 

.4006  2.4960 

.4210  2.3750 

.4417 

2.2637 

.4628  2.1609 

10 

51 

09   6256 

10   4939 

14   373] 

21 

2620 

31   1592 

9 

52 

12   6233 

13   4918 

17   3712 

24 

2602 

35   1576 

8 

53 

15   6210 

17   4897 

21   3693 

28 

2584 

38   1560 

7 

54 

19   6187 

20   4876 

24   3673 

31 

2566 

42   1543 

6 

55 

.3822  2.6165 

.4023  2.4855 

.4228  2.3654 

.4435 

2.2549 

.4645  2.1527 

5 

56 

25   6142 

27   4834 

31   3635 

38 

2531 

49   1510 

4 

57 
58 

29   6119 
32   6096 

30   4813 

34  3616 

42 
45 

2513 
2496 

52   1494 
56   1478 

3 
2 

33   4792 

38  3597 

59 

35   6074 

37   4772 

41   3578 

49 

2478 

60   1461 

1 

60 

.3839  2.6051 

.4040  2.4751 

.4245  2.3559 

.4452 

2.2460 

.4663  2.1445 

0 

t 

cot   tan 

cot   tan 

cot   tan 

cot 

tan 

cot   tan 

69° 

68" 

67° 

66° 

65° 

Z. 

85 


Natural  Sines  and  Cosines 


1 

25^ 

26^ 

27 

o 

28 

f 

29 

-■ 

/ 

sin 

COS 

sin 

cos 

sin 

COS 

sin 

COS 

sin 

COS 

0 

.4226 

.9063 

.4384 

.8988 

.4540 

.8910 

.4695 

.8829 

.4848 

.8746 

60 

1 

29 

62 

86 

87 

42 

09 

97 

28 

51 

45 

59 

2 

31 

61 

89 

85 

45 

07 

4700 

27 

53 

43 

58 

3 

34 

59 

92 

84 

48 

06 

02 

25 

56 

42 

57 

4 

37 

58 

94 

83 

50 

05 

05 

24 

58 

41 

56 

5 

.4239 

.9057 

.4397 

.8982 

.4553 

.8903 

.4708 

.8823 

.4861 

.8739 

55 

6 

42 

56 

99 

80 

55 

02 

10 

21 

63 

38 

54 

7 

45 

54 

4402 

79 

58 

01 

13 

20 

66 

36 

53 

8 

47 

53 

05 

78 

61 

8899 

15 

19 

68 

35 

52 

9 

iO 

52 

07 

76 

63 

98 

18 

17 

71 

33 

51 

10 

.4253 

.9051 

.4410 

.8975 

.4566 

.8897 

.4720 

.8816 

.4874 

.8732 

50 

11 

55 

50 

12 

74 

68 

95 

23 

14 

76 

31 

49 

12 

58 

48 

15 

73 

71 

94 

26 

13 

79 

29 

48 

13 

60 

47 

18 

71 

74 

93 

28 

12 

81 

28 

47 

14 

63 

46 

20 

70 

76 

92 

31 

10 

84 

26 

46 

15 

.4266 

.9045 

.4423 

.8969 

.4579 

.8890 

.4733 

.8809 

.4886 

.8725 

45 

16 

68 

43 

25 

67 

81 

89 

36 

08 

89 

24 

44 

17 

71 

42 

28 

66 

84 

88 

38 

06 

91 

22 

43 

18 

74 

41 

31 

65 

86 

86 

41 

05 

94 

21 

42 

19 

76 

40 

33 

64 

89 

85 

43 

03 

96 

19 

41 

20 

.4279 

.9038 

.4436 

.8962 

.4592 

.8884 

.4746 

.8802 

.4899 

.8718 

40 

21 

81 

37 

39 

61 

94 

82 

49 

01 

4901 

16 

39 

22 

84 

36 

41 

60 

97 

81 

51 

8799 

04 

15 

38 

23 

87 

35 

44 

58 

99 

79 

54 

98 

07 

14 

37 

24 

89 

33 

46 

57 

4602 

78 

56 

% 

09 

12 

36 

25 

.4292 

.9032 

.4449 

.8956 

.4605 

.8877 

.4759 

.8795 

.4912 

.8711 

35 

26 

95 

31 

52 

55 

07 

75 

61 

94 

14 

09 

34 

27 

97 

30 

54 

53 

10 

74 

64 

92 

17 

08 

33 

28 

4300 

28 

57 

52 

12 

73 

66 

91 

19 

06 

32 

29 

02 

27 

59 

51 

15 

71 

69 

90 

22 

05 

31 

30 

.4305 

.9026 

.4462 

.8949 

.4617 

.8870 

.4772 

.8788 

.4924 

.8704 

30 

31 

08 

25 

65 

48 

20 

69 

74 

87 

27 

02 

29 

32 

10 

23 

67 

47 

23 

67 

77 

85 

29 

01 

28 

33 

13 

22 

70 

45 

25 

66 

79 

84 

32 

8699 

27 

34 

16 

21 

72 

44 

28 

65 

82 

83 

34 

98 

26 

35 

.4318 

.9020 

.4475 

.8943 

.4630 

.8863 

.4784 

.8781 

.4937 

.86% 

25 

36 

21 

18 

78 

42 

33 

62 

87 

80 

39 

95 

24 

37 

23 

17 

80 

40 

36 

61 

89 

78 

42 

94 

23 

38 

26 

16 

83 

39 

38 

59 

92 

77 

44 

92 

22 

39 

29 

15 

85 

38 

41 

58 

95 

76 

47 

91 

21 

40 

.4331 

.9013 

.4488 

.8936 

.4643 

.8857 

.4797 

.8774 

.4950 

.8689 

20 

41 

34 

12 

91 

35 

46 

55 

4800 

73 

52 

88 

19 

42 

37 

11 

93 

34 

48 

54 

02 

71 

55 

86 

18 

43 

39 

10 

96 

32 

51 

53 

05 

70 

57 

85 

17 

44 

42 

08 

98 

31 

54 

51 

07 

69 

60 

83 

16 

45 

.4344 

.9007 

.4501 

.8930 

.4656 

.8850 

.4810 

^767 

.4%2 

.8682 

15 

46 

47 

06 

04 

28 

59 

49 

12 

66 

65 

81 

14 

47 

50 

04 

06 

27 

61 

47 

15 

64 

67 

79 

13 

48 

52 

03 

09 

26 

64 

46 

18 

63 

70 

78 

12 

49 

55 

02 

11 

25 

66 

44 

20 

62 

72 

76 

11 

50 

.4358 

.9001 

.4514 

.8923 

.4669 

.8843 

.4823 

.8760 

.4975 

.8675 

10 

51 

60 

8999 

17 

22 

72 

42 

25 

59 

77 

73 

9 

52 

63 

98 

19 

21 

74 

40 

28 

57 

80 

72 

8 

53 

65 

97 

22 

19 

77 

39 

30 

56 

82 

70 

7 

54 

68 

96 

24 

18 

79 

38 

33 

55 

85 

69 

6 

55 

.4371 

.8994 

.4527 

.8917 

.4682 

.8836 

.4835 

.8753 

.4987 

.8668 

5 

56 

73 

93 

30 

IS 

84 

35 

38 

52 

90 

66 

4 

57 

76 

92 

32 

14 

87 

34 

40 

50 

92 

65 

3 

58 

78 

90 

35 

13 

90 

32 

43 

49 

95 

63 

2 

59 

81 

89 

37 

11 

92 

31 

46 

48 

97 

62 

1 

60 

.4384 

.8988 

.4540 

.8910 

.4695 

.8829 

.4848 

.8746 

.5000 

.8660 

0 

COS 

sin 

cos 

sin 

cos 

sin 

COS 

sin 

COS 

sin 

1 

64° 

63° 

62° 

61° 

60 

1 

86 


Natural  Tangents  and  Cotangents 


/ 

25° 

26° 

27° 

28° 

29° 

/ 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

0 

.4663 

2.1445 

.4877 

2.0503 

.5095 

1.9626 

.5317 

1.8807 

.5543 

1.8040 

60 

1 

67 

1429 

81 

0488 

99 

9612 

21 

8794 

47 

8028 

59 

2 

70 

1413 

85 

0473 

5103 

9598 

25 

8781 

51 

8016 

58 

3 

74 

1396 

88 

0458 

06 

9584 

28 

8768 

55 

8003 

57 

4 

77 

1380 

92 

0443 

10 

9570 

32 

875i 

58 

7991 

56 

5 

.4681 

2.1364 

.4895 

2.0428 

.5114 

1.9556 

.5336 

1.8741 

.5562 

1.7979 

55 

6 

84 

1348 

99 

0413 

17 

9542 

40 

8728 

66 

7966 

54 

7 

88 

1332 

4903 

0398 

21 

9528 

43 

8715 

70 

7954 

53 

8 

91 

1315 

06 

0383 

25 

9514 

47 

8702 

74 

7942 

52 

9 

95 

1299 

10 

0368 

28 

9500 

51 

8689 

77 

7930 

51 

10 

.4699 

2.1283 

.4913 

2.0353 

.5132 

1.9486 

.5354 

1.8676 

.5.581 

1.7917 

50 

11 

4702 

1267 

17 

0338 

36 

9472 

58 

8663 

85 

7905 

49 

12 

06 

1251 

21 

0323 

39 

9458 

62 

8650 

89 

7893 

48 

13 

09 

1235 

24 

0308 

43 

9444 

66 

8637 

93 

7881 

47 

14 

13 

1219 

28 

0293 

47 

9430 

69 

8624 

96 

7868 

46 

15 

.4716 

2.1203 

.4931 

2.0278 

.5150 

1.9416 

.5373 

1.8611 

.5600 

1.7856 

45 

16 

20 

1187 

35 

0263 

54 

9402 

77 

8598 

04 

7844 

44 

17 

23 

1171 

39 

0248 

58 

9388 

81 

8585 

08 

7832 

43 

18 

27 

1155 

42 

0233 

61 

9375 

84 

8572 

12 

7820 

42 

19 

31 

1139 

46 

0219 

65 

9361 

88 

8559 

16 

7808 

41 

20 

.4734 

2.1123 

.4950 

2.0204 

.5169 

1.9347 

.5392 

1.8546 

.5619 

1.7796 

40 

21 

38 

1107 

53 

0189 

72 

9333 

96 

8533 

23 

7783 

39 

22 

41 

1092 

57 

0174 

76 

9319 

99 

8520 

27 

7771 

38 

23 

45 

1076 

60 

0160 

80 

9306 

5403 

8507 

31 

7759 

37 

24 

48 

1060 

64 

0145 

84 

9292 

07 

8495 

35 

7747 

36 

25 

.4752 

2.1044 

.4968 

2.0130 

.5187 

1.9278 

.5411 

1.8482 

.5639 

1.7735 

35 

26 

55 

1028 

71 

0115 

91 

9265 

li 

8469 

42 

7723 

34 

27 

59 

1013 

75 

0101 

95 

9251 

18 

8456 

46 

7711 

33 

28 

63 

0997 

79 

0086 

98 

9237 

22 

8443 

50 

7699 

32 

29 

66 

0981 

82 

0072 

5202 

9223 

26 

8430 

54 

7687 

31 

30 

.4770 

2.0965 

.4986 

2.0057 

.5206 

1.9210 

.5430 

1.8418 

.5658 

1.7675 

30 

31 

73 

0950 

89 

0042 

09 

9196 

33 

8405 

62 

7663 

29 

32 

77 

0934 

93 

0028 

13 

9183 

37 

8392 

65 

7651 

28 

33 

80 

0918 

97 

0013 

17 

9169 

41 

8379 

69 

7639 

27 

34 

84 

0903 

5000 

1.9999 

20 

9155 

45 

8367 

73 

7627 

26 

35 

.4788 

2.0887 

.5004 

1.9984 

.5224 

1.9142 

.5448 

1.8354 

.5677 

1.7615 

25 

36 

91 

0872 

08 

9970 

28 

9128 

52 

8341 

81 

7603 

24 

37 

95 

0856 

11 

9955 

32 

9115 

56 

8329 

85 

7591 

23 

38 

98 

0840 

15 

9941 

35 

9101 

60 

8316 

88 

7579 

22 

39 

4802 

0825 

19 

9926 

39 

9088 

64 

8303 

92 

7567 

21 

40 

.4806 

2.0809 

.5022 

1.9912 

.5243 

1.9074 

.5467 

1.8291 

.5696 

1.7556 

20 

41 

09 

0794 

26 

9897 

46 

9061 

71 

8278 

5700 

7544 

19 

42 

13 

0778 

29 

9883 

50 

9047 

75 

8265 

04 

7532 

18 

43 

16 

0763 

33 

9868 

54 

9034 

79 

8253 

08 

7520 

17 

44 

20 

0748 

37 

9854 

58 

9020 

82 

8240 

12 

7508 

16 

46 

.4823 

2.0732 

.5040  1.9840 

.5261 

1.9007 

.5486 

1.8228 

.5715 

1.7496 

15 

46 

27 

0717 

44 

9825 

65 

8993 

90 

8215 

19 

7485 

14 

47 

31 

0701 

48 

9811 

69 

8980 

94 

8202 

23 

7473 

13 

48 

34 

0686 

51 

9797 

72 

8967 

98 

8190 

27 

7461 

12 

49 

38 

0671 

55 

9782 

76 

8953 

5501 

8177 

31 

7449 

11 

50 

.4841 

2.0655 

.5059 

1.9768 

.5280 

1.8940 

.5505 

1.8165 

.5735 

1.7437 

10 

51 

45 

0640 

62 

9754 

84 

8927 

09 

8152 

39 

7426 

9 

52 

49 

0625 

66 

9740 

87 

8913 

13 

8140 

43 

7414 

8 

53 

52 

0609 

70 

9725 

91 

8900 

17 

8127 

46 

7402 

7 

54 

56 

0594 

73 

9711 

95 

8887 

20 

8115 

50 

7391 

6 

55 

.4859 

2.0579 

.5077 

1.9697 

.5298 

1.8873 

.5524 

1.8103 

.5754 

1.7379 

5 

56 

63 

0564 

81 

9683 

5302 

8860 

28 

8090 

58 

7367 

4 

57 

58 

67 
70 

0549 
0533 

84 
88 

9669 

06 

8847 
8834 

32 

35 

8078 
8065 

62 
66 

7355 
7344 

3 
2 

9654 

10 

59 

74 

0518 

92 

9640 

13 

8820 

39 

8053 

70 

7332' 

1 

60 

.4877 

2.0503 

.5095 

1.9626 

.5317 

1.8807 

.5543 

1.8040 

.5774 

1.7321 

0 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

/ 

64° 

63° 

62° 

61° 

60" 

/ 

87 


Natural  Sixes  and  Cosixes 


1 

30° 

31° 

32"^ 

33° 

34 

^ 

/ 

sin 

cos 

sin 

cos 

sin 

COS 

sin 

cos 

sin 

cos 

0 

.5000 

.8660 

.5150 

.8572 

.5299 

.8480 

.5446 

.8387 

.5592 

.8290 

60 

1 

03 

59 

53 

70 

5302 

79 

49 

85 

94 

89 

59 

2 

05 

57 

55 

69 

04 

77 

51 

84 

97 

87 

58 

3 

08 

56 

58 

67 

07 

76 

54 

82 

99 

85 

57 

4 

10 

54 

60 

66 

09 

74 

56 

80 

5602 

84 

56 

5 

.5013 

.86.53 

.5163 

.8564 

.5312 

.8473 

.5459 

.8379 

.5604 

.8282 

56 

6 

15 

52 

65 

63 

14 

71 

61 

77 

06 

81 

54 

7 

18 

50 

68 

61 

16 

70 

63 

76 

09 

79 

53 

8 

20 

49 

70 

60 

19 

68 

66 

74 

11 

77 

52 

9 

23 

47 

73 

58 

21 

67 

68 

72 

14 

76 

51 

10 

.5025 

.8646 

.5175 

.8557 

.5324 

.8465 

.5471 

.8371 

.5616 

.8274 

50 

11 

28 

44 

78 

55 

26 

63 

73 

69 

18 

72 

49 

12 

30 

43 

80 

54 

29 

62 

76 

68 

21 

71 

48 

13 

33 

41 

83 

52 

31 

60 

78 

66 

23 

69 

47 

14 

35 

40 

85 

51 

34 

59 

80 

64 

26 

68 

46 

16 

.5038 

.8638 

.5188 

.8549 

.5336 

.8457 

.5483 

.8363 

.5628 

.8266 

45 

16 

40 

37 

90 

48 

39 

56 

85 

61 

30 

64 

44 

17 

43 

35 

93 

46 

41 

54 

88 

60 

33 

63 

43 

18 

45 

34 

95 

4i 

44 

53 

90 

58 

35 

61 

42 

19 

48 

32 

98 

43 

46 

51 

93 

56 

38 

59 

41 

20 

.5050 

.8631 

.5200 

.8542 

.5348 

.8450 

.5495 

.8355. 

.5640 

.8258 

40 

21 

53 

30 

03 

40 

51 

48 

98 

53 

42 

56 

39 

22 

55 

28 

05 

39 

53 

46 

5i00 

52 

45 

54 

38 

23 

58 

27 

08 

37 

56 

45 

02 

50 

47 

53 

37 

24 

60 

25 

10 

36 

58 

43 

05 

48 

50 

51 

36 

25 

.5063 

.8624 

.5213 

.8534 

.5361 

.8442 

.5507 

.8347 

.5652 

.8249 

35 

26 

65 

22 

15 

32 

63 

40 

10 

45 

54 

48 

34 

27 

68 

21 

18 

31 

66 

39 

12 

44 

57 

46 

33 

28 

70 

19 

20 

29 

68 

37 

15 

42 

59 

45 

32 

29 

73 

18 

23 

28 

71 

35 

17 

40 

62 

43 

31 

30 

.5075 

.8616 

.5225 

.8526 

.5373 

.8434 

.5519 

.8339 

.5664 

.8241 

30 

31 

78 

15 

27 

25 

75 

32 

22 

37 

66 

40 

29 

32 

80 

13 

30 

23 

78 

31 

24 

36 

69 

38 

28 

33 

83 

12 

32 

22 

80 

29 

27 

34 

71 

36 

27 

34 

85 

10 

35 

20 

83 

28 

29 

32 

74 

35 

26 

36 

.5088 

.8609 

.5237 

.8519 

.5385 

.8426 

.5531 

.8331 

.5676 

.8233 

25 

36 

90 

07 

40 

17 

88 

25 

34 

29 

78 

31 

2A 

37 

93 

06 

42 

16 

90 

23 

36 

28 

81 

30 

23 

38 

95 

04 

45 

14 

93 

21 

39 

26 

83 

28 

22 

39 

98 

03 

47 

13 

95 

20 

41 

24 

86 

26 

21 

40 

.5100 

.8601 

.5250 

.8511 

.5398 

.8418 

.5544 

.8323 

.5688 

.8225 

20 

41 

03 

00 

52 

10 

5400 

17 

46 

21 

90 

23 

19 

42 

05 

8599 

55 

08 

02 

15 

48 

20 

93 

21 

18 

43 

08 

97 

57 

07 

05 

14 

51 

18 

95 

20 

17 

44 

10 

96 

60 

05 

07 

12 

53 

16 

98 

18 

16 

46 

.5113 

.8594 

.5262 

.8504 

.5410 

.8410 

.5556 

.8315 

.5700 

.8216 

15 

46 

15 

93 

65 

02 

12 

09 

58 

13 

02 

15 

14 

47 

18 

91 

67 

00 

15 

07 

61 

11 

05 

13 

13 

48 

20 

90 

70 

8499 

17 

06 

63 

10 

07 

11 

12 

49 

23 

88 

72 

97 

20 

04 

65 

08 

10 

10 

11 

50 

.5125 

.8587 

.5275 

.8496 

.5422 

.8403 

.5568 

.8307 

.5712 

.8208 

10 

51 

28 

85 

77 

94 

24 

01 

70 

05 

14 

07 

9 

52 

30 

84 

79 

93 

27 

8399 

73 

03 

17 

05 

8 

53 

33 

82 

82 

91 

29 

98 

75 

02 

19 

03 

7 

54 

35 

81 

84 

90 

32 

96 

77 

00 

21 

02 

6 

55 

.5138 

.8579 

.5287 

.8488 

.5434 

.8395 

.5580 

.8298 

.5724 

.8200 

5 

56 

40 

78 

89 

87 

37 

93 

82 

97 

26 

8198 

4 

57 

43 

76 

92 

85 

39 

91 

85 

95 

29 

97 

3 

58 

45 

75 

94 

84 

42 

90 

87 

94 

31 

95 

2 

59 

48 

73 

97 

82 

44 

88 

90 

92 

33 

93 

1 

60 

.5150 

.8572 

.5299 

.8480 

.5446 

.8387 

.5592 

.8290 

.5736 

.8192 

0 

cos 

sin 

COS 

sin 

COS 

sin 

COS 

sin 

cos 

sin 

t 

5S 

-" 

58 

° 

57 

o 

56 

o 

55 

3 

/ 

88 


Natural  Tangents  and  Cotangents 


/ 

30° 

31° 

32° 

33° 

34° 

/ 

tan 

cot 

tan   cot 

tan 

cot 

tan 

cot 

tan 

cot 

0 

.5774 

1.7321 

.6009  1.6643 

.6249 

1.6003 

.6494 

1.5399 

.6745 

1.4826 

60 

1 

77 

7309 

13   6632 

53 

5993 

98 

5389 

49 

4816 

59 

2 

81 

7297 

17   6621 

57 

5983 

6502 

5379 

54 

4807 

58 

3 

85 

7286 

20   6610 

61 

5972 

06 

5369 

58 

4798 

57 

4 

89 

7274 

24   6599 

65 

5962 

11 

5359 

62 

4788 

56 

6 

.5793 

1.7262 

.6028  1.6588 

.6269 

1.5952 

.6515 

1..S350 

.6766 

1.4779 

55 

6 

97 

7251 

32   6577 

73 

5941 

19 

5340 

71 

4770 

54 

7 

5801 

7239 

36   6566 

77 

5931 

23 

5330 

75 

4761 

53 

8 

05 

7228 

40   6555 

81 

5921 

27 

5320 

79 

4751 

52 

9 

08 

7216 

44   6545 

85 

5911 

31 

5311 

83 

4742 

51 

10 

.5812 

1.7205 

.6048  1.6534 

.6289 

1.5900 

.6536 

1.5301 

.6787 

1.4733 

50 

11 

16 

7193 

52   6523 

93 

5890 

40 

5291 

92 

4724 

49 

12 

20 

7182 

56   6512 

97 

5880 

44 

5282 

96 

4715 

48 

13 

24 

7170 

60   6501 

6301 

5869 

48 

5272 

6800 

4705 

47 

14 

28 

7159 

64   6490 

05 

5859 

52 

5262 

05 

4696 

46 

15 

.5832 

1.7147 

.6068  1.6479 

.6310 

1.5849 

.6556 

1.5253 

.6809 

1.4687 

45 

16 

36 

7136 

72   6469 

14 

5839 

60 

5243 

13 

4678 

44 

17 

40 

7124 

76   6458 

18 

5829 

65 

5233 

17 

4669 

43 

18 

44 

7113 

80   6447 

22 

5818 

69 

5224 

22 

4659 

42 

19 

47 

7102 

84   6436 

26 

5808 

73 

5214 

26 

4650 

41 

20 

.5851 

1.7090 

.6088  1.6426 

.6330 

1.5798 

.6577 

1.5204 

.6830 

1.4641 

40 

21 

55 

7079 

92   6415 

34 

5788 

81 

5195 

34 

4632 

39 

22 

59 

7067 

96   6404 

38 

5778 

85 

5185 

39 

4623 

38 

23 

63 

7056 

6100   6393 

42 

5768 

90 

5175 

43 

4614 

37 

24 

67 

7045 

04   6383 

46 

5757 

94 

5166 

47 

4605 

36 

25 

.5871 

1.7033 

.6108  1.6372 

.6350 

1.5747 

.6598 

1.5156 

.6851 

1.4596 

35 

26 

75 

7022 

12   6361 

54 

5737 

6602 

5147 

56 

4586 

34 

27 

79 

7011, 

16   6351 

58 

5727 

06 

5137 

60 

4577 

Zl 

28 

83 

6999 

20   6340 

63 

5717 

10 

5127 

64 

4568 

32 

29 

87 

6988 

24   6329 

67 

5707 

15 

5118 

69 

4559 

31 

30 

.5890 

1.6977 

.6128  1.6319 

.6371 

1.5697 

.6619 

1.5108 

.6873 

1.4550 

30 

31 

94 

6965 

32   6308 

75 

5687 

23 

5099 

77 

4541 

29 

32 

98 

6954 

36   6297 

79 

5677 

27 

5089 

81 

4532 

28 

33 

5902 

6943 

40   6287 

83 

5667 

31 

5080 

86 

4523 

27 

34 

06 

6932 

44   6276 

87 

5657 

36 

5070 

90 

4514 

26 

35 

.5910 

1.6920 

.6148  1.6265 

.6391 

1.5647 

.6640 

1.5061 

.6894 

1.4505 

25 

36 

14 

6909 

52   6255 

95 

5637 

44 

5051 

99 

4496 

24 

37 

18 

6898 

56   6244 

99 

5627 

48 

5042 

6903 

4487 

23 

38 

22 

6887 

60   6234 

6403 

5617 

52 

5032 

07 

4478 

22 

39 

26 

6875 

64   6223 

08 

5607 

57 

5023 

11 

4469 

21 

40 

.5930 

1.6864 

.6168  1.6212 

.6412 

1.5597 

.6661 

1.5013 

.6916 

1.4460 

20 

41 

34 

6853 

72   6202 

16 

5587 

65 

5004 

20 

4451 

19 

42 

38 

6842 

76   6191 

20 

5577 

69 

4994 

24 

4442 

18 

43 

42 

6831 

80   6181 

24 

5567 

73 

4985 

29 

4433 

17 

44 

45 

6820 

84   6170 

28 

5557 

78 

4975 

IZ 

4424 

16 

45 

.5949 

1.6808 

.6188  1.6160 

.6432 

1.5547 

.6682 

1.4966 

.6937 

1.4415 

15 

46 

53 

6797 

92   6149 

36 

5537 

86 

4957 

42 

4406 

14 

47 

57 

6786 

96   6139 

40 

5527 

90 

4947 

46 

4397 

13 

48 

61 

6775 

6200   6128 

45 

5517 

94 

4938 

50 

4388 

12 

49 

65 

6764 

04   6118 

49 

5507 

99 

4928 

54 

4379 

11 

50 

.5969 

1.6753 

.6208  1.6107 

.6453 

1.5497 

.6703 

1.4919 

.6959 

1.4370 

10 

51 

73 

6742 

12   6097 

57 

5487 

07 

4910 

63 

4361 

9 

52 

77 

6731 

16   6087 

61 

5477 

11 

4900 

67 

4352 

8 

53 

81 

6720 

20   6076 

65 

5468 

15 

4891 

72 

4344 

7 

54 

85 

6709 

24   6066 

69 

5458 

20 

4882 

76 

4335 

6 

55 

.5989 

1.6698 

.6228  1.6055 

.6473 

1.5448 

.6724 

1.4872 

.6980 

1.4326 

5 

56 

93 

6687 

33   6045 

78 

5438 

28 

4863 

85 

4317 

4 

57 
58 

97 
6001 

6676 
6665 

37   6034 
41   6024 

82 

5428 

32 
37 

4854 
4844 

89 
93 

4308 
4299 

3 

2 

86 

5418 

59 

05 

6654 

45   6014 

90 

5408 

41 

4835 

98 

4290 

1 

60 

.6009 

1.6643 

.6249  1.6003 

.6494 

1.5399 

.6745 

1.4826 

.7002 

1.4281 

0 

cot 

tan 

cot   tan 

cot 

tan 

cot 

tan 

cot 

tan 

1 

59° 

58° 

57° 

56° 

55°     1 

1 

89 


Natural  Sixes  axd  Cosixes 


/ 

35 

o 

36 

o 

37 

0 

38 

o 

39°    1 

/ 

sin 

COS 

sin 

COS 

sin 

COS 

sin 

COS 

sin 

COS 

0 

.5736 

.8192 

.5878 

.8090 

.6018 

.7986 

.6157 

.7880 

.6293 

.7771 

60 

1 

38 

90 

80 

88 

20 

85 

59 

78 

95 

70 

59 

2 

41 

88 

83 

87 

23 

83 

61 

77 

98 

68 

58 

3 

43 

87 

85 

85 

25 

81 

63 

75 

6300 

66 

57 

4 

45 

85 

87 

83 

27 

79 

66 

73 

02 

64 

56 

6 

.5748 

.8183 

.5890 

.8082 

.6030 

.7978 

.6168 

.7871 

.6305 

.7762 

55 

6 

50 

81 

92 

80 

32 

76 

70 

69 

07 

60 

54 

7 

52 

80 

94 

78 

34 

74 

73 

68 

09 

59 

53 

8 

55 

78 

97 

76 

37 

72 

75 

66 

11 

57 

52 

9 

57 

76 

99 

75 

39 

71 

77 

64 

14 

55 

51 

10 

.5760 

.8175 

.5901 

.8073 

.6041 

.7%9 

.6180 

.7862 

.6316 

.7753 

60 

11 

62 

73 

04 

71 

44 

67 

82 

60 

18 

51 

49 

12 

64 

71 

06 

70 

46 

65 

84 

59 

20 

49 

48 

13 

67 

70 

08 

68 

48 

64 

86 

57 

23 

48 

47 

14 

69 

68 

11 

66 

51 

62 

89 

55 

25 

46 

46 

16 

.5771 

.8166 

.5913 

.8064 

.6053 

.7960 

.6191 

.7853 

.6327 

.7744 

46 

16 

74 

65 

15 

63 

55 

58 

93 

51 

29 

42 

44 

17 

76 

63 

18 

61 

58 

56 

% 

50 

32 

40 

43 

18 

79 

61 

20 

59 

60 

■55 

98 

48 

34 

38 

42 

19 

81 

60 

22 

58 

62 

53 

6200 

46 

36 

37 

41 

20 

.5783 

.8158 

.5925 

.8056 

.6065 

.7951 

.6202 

.7844 

.6338 

.7735 

40 

21 

86 

56 

27 

54 

67 

49 

05 

42 

41 

33 

39 

22 

88 

55 

30 

52 

69 

48 

07 

41 

43 

31 

38 

23 

90 

53 

32 

51 

71 

46 

09 

39 

45 

29 

37 

24 

93 

51 

34 

49 

74 

44 

11 

37 

47 

27 

36 

26 

.5795 

.8150 

.5937 

.8047 

.6076 

.7942 

.6214 

.7835 

.6350 

.7725 

36 

26 

98 

48 

39 

45 

78 

41 

16 

33 

52 

24 

34 

27 

5800 

46 

41. 

44 

81 

39 

18 

32 

54 

22 

33 

28 

02 

45 

44 

42 

83 

37 

21 

30 

56 

20 

32 

29 

05 

43 

46 

40 

85 

35 

23 

28 

59 

18 

31 

30 

.5807 

.8141 

.5948 

.8039 

.6088 

.7934 

.6225 

.7826 

.6361 

.7716 

30 

31 

09 

39 

51 

37 

90 

32 

27 

24 

63 

14 

29 

32 

12 

38 

53 

35 

92 

30 

30 

22 

65 

13 

28 

33 

14 

36 

55 

33 

95 

28 

32 

21 

68 

11 

27 

34 

16 

34 

58 

32 

97 

26 

34 

19 

70 

09 

26 

36 

.5819 

.8133 

.5960 

.8030 

.6099 

.7925 

.6237 

.7817 

.6372 

.7707 

26 

36 

21 

31 

62 

28 

6101 

23 

39 

15 

74 

05 

24 

37 

24 

29 

65 

26 

04 

21 

41 

13 

76 

03 

23 

38 

26 

28 

67 

25 

06 

19 

43 

12 

79 

01 

22 

39 

28 

26 

69 

23 

08 

18 

46 

10 

81 

00 

21 

40 

.5831 

.8124 

.5972 

.8021 

.6111 

.7916 

.6248 

.7808 

.6383 

.7698 

20 

41 

33 

23 

74 

19 

13 

14 

50 

06 

85 

96 

19 

42 

35 

•  21 

76 

18 

15 

12 

52 

04 

88 

94 

18 

43 

38 

19 

79 

16 

18 

10 

55 

02 

90 

92 

17 

44 

40 

17 

81 

14 

20 

09 

57 

01 

92 

90 

16 

46 

.5842 

.8116 

.5983 

.8013 

.6122 

.7907 

.6259 

.7799 

.6394 

.7688 

16 

46 

4i 

14 

86 

11 

24 

05 

62 

97 

97 

87 

14 

47 

47 

12 

88 

09 

27 

03 

64 

95 

99 

85 

13 

48 

50 

11 

90 

07 

29 

02 

66 

93 

6401 

83 

12 

49 

52 

09 

93 

06 

31 

00 

68 

92 

03 

81 

11 

60 

.5854 

.8107 

.5995 

.8004 

.6134 

.7898 

.6271 

.7790 

.6406 

.7679 

10 

51 

57 

06 

97 

02 

36 

96 

73 

88 

08 

77 

9 

52 

59 

04 

6000 

00 

38 

94 

75 

86 

10 

75 

8 

53 

61 

02 

02 

7999 

41 

93 

77 

84 

12 

74 

7 

54 

64 

00 

04 

97 

43 

91 

80 

82 

14 

72 

6 

65 

.5866 

.8099 

.6007 

.7995 

.6145 

.7889 

.6282 

.7781 

.6417 

.7670 

6 

56 

68 

97 

09 

93 

47 

87 

84 

79 

19 

68 

4 

57 

71 

95 

11 

92 

50 

85 

86 

77 

21 

66 

3 

58 

73 

94 

14 

90 

52 

84 

89 

75 

23 

64 

2 

59 

75 

92 

16 

88 

54 

82 

91 

73 

26 

62 

1 

60 

.5878 

.8090 

.6018 

.7986 

.6157 

.7880 

.6293 

.7771 

.6428 

.7660 

0 

cos 

sin 

COS 

sin 

COS 

sin 

COS 

sin 

COS 

sin 

L 

54° 

53° 

52° 

51° 

50^ 

1 

90 


Natural  Tangents  and  Cotangents 


1 

35° 

36° 

37° 

38° 

39° 

/ 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan   cot 

0 

.7002 

1.4281 

.7265 

1.3764 

.7536 

1.3270 

.7813 

1.2799 

.8098  1.2349 

60 

1 

06 

4273 

70 

3755 

40 

3262 

18 

2792 

8103   2342 

59 

2 

11 

4264 

74 

3747 

45 

3254 

22 

2784 

07   2334 

58 

3 

15 

4255 

79 

3739 

49 

3246 

27 

2776 

12   2327 

57 

4 

19 

4246 

83 

3730 

54 

3238 

32 

2769 

17   2320 

56 

5 

.7024 

1.4237 

.7288 

1.3722 

.7558 

1.3230 

.7836 

1.2761 

.8122  1.2312 

55 

6 

28 

4229 

92 

3713 

63 

3222 

41 

2753 

27   2305 

54 

7 

32 

4220 

97 

3705 

68 

3214 

46 

2746 

32   2298 

53 

8 

37 

4211 

7301 

3697 

72 

3206 

50 

2738 

36   2290 

52 

9 

41 

4202 

06 

3688 

77 

3198 

55 

2731 

41   2283 

51 

10 

.7046 

1.4193 

.7310 

1.3680 

.7581 

1.3190 

.7860 

1.2723 

.8146  1.2276 

50 

11 

■  50 

4185 

14 

3672 

86 

3182 

65 

2715 

51   2268 

49 

12 

54 

4176 

19 

3663 

90 

3175 

69 

2708 

56   2261 

48 

13 

59 

4167 

23 

3655 

95 

3167 

74 

2700 

61   2254 

47 

14 

63 

4158 

28 

3647 

7600 

3159 

79 

2693 

65   2247 

46 

15 

.7067 

1.4150 

.7332 

1.3638 

.7604 

1.3151 

.7883 

1.2685 

.8170  1.2239 

45 

16 

72 

4141 

37 

3630 

09 

3143 

88 

2677 

75   2232 

44 

17 

76 

4132 

41 

3622 

13 

3135 

93 

2670 

80   2225 

43 

18 

80 

4124 

46 

3613 

IS 

3127 

98 

2662 

85   2218 

42 

19 

85 

4115 

50 

3605 

23 

3119 

7902 

2655 

90   2210 

41 

20 

.7089 

1.4106 

.7355 

1.3597 

.7627 

1.3111 

.7907 

1.2647 

.8195  1.2203 

40 

21 

94 

4097 

59 

3588 

32 

3103 

12 

2640 

99   2196 

39 

22 

98 

4089 

64 

3580 

36 

3095 

16 

2632 

8204   2189 

38 

23 

7102 

4080 

68 

3572 

41 

3087 

21 

2624 

09   2181 

37 

24 

07 

4071 

73 

3564 

46 

3079 

26 

2617 

14   2174 

36 

25 

.7111 

1.4063 

.7377 

1.3555 

.7650 

1.3072 

.7931 

1.2609 

.8219.  1.2167 

35 

26 

15 

4054 

82 

3547 

55 

3064 

35 

2602 

24   2160 

34 

27 

20 

4045 

86 

3539 

59 

3056 

40 

2594 

29   2153 

33 

28 

24 

4037 

91 

3531 

64 

3048 

45 

2587 

34   2145 

32 

29 

29 

4028 

95 

3522 

69 

3040 

50 

2579 

38   2138 

31 

30 

.7133 

1.4019 

.7400 

1.3514 

.7673 

1.3032 

.7954 

1.2572 

.8243  1.2131 

30 

31 

37 

4011 

04 

3506 

78 

3024 

59 

2564 

48   2124 

29 

32 

42 

4002 

09 

3498 

83 

3017 

64 

2557 

53   2117 

28 

33 

46 

3994 

13 

3490 

87 

3009 

69 

2549 

58   2109 

27 

34 

51 

3985 

18 

3481 

92 

3001 

73 

2542 

63   2102 

26 

35 

.7155 

1.3976 

.7422 

1.3473 

.7696 

1.2993 

.7978 

1.2534 

.8268  1.2095 

25 

36 

59 

3968 

27 

3465 

7701 

2985 

83 

2527 

73   2088 

24 

37 

64 

3959 

31 

3457 

06 

2977 

88 

2519 

78   2081 

23 

38 

68 

3951 

36 

3449 

10 

2970 

92 

2512 

83   2074 

22 

39 

73 

3942 

40 

3440 

15 

2962 

97 

2504 

87   2066 

21 

40 

.7177 

1.3934 

.7445 

1.3432 

.7720 

1.2954 

.8002 

1.2497 

.8292  1.2059 

20 

41 

81 

3925 

49 

3424 

24 

2946 

07 

2489 

97   2052 

19 

42 

86 

3916 

54 

3416 

29 

2938 

12 

2482 

8302   2045 

18 

43 

90 

3908 

58 

3408 

34 

2931 

16 

2475 

07   2038 

17 

44 

95 

3899 

63 

3400 

38 

2923 

21 

2467 

12   2031 

16 

45 

.7199 

1.3891 

.7467 

1.3392 

.7743 

1.2915 

.8026 

1.2460 

.8317  1.2024 

15 

46 

7203 

3882 

72 

3384 

47 

2907 

31 

2452 

22   2017 

14 

47 

08 

3874 

76 

3375 

52 

2900 

35 

2445 

27   2009 

13 

48 

12 

3865 

81 

3367 

57 

2892 

40 

2437 

32   2002 

12 

49 

17 

3857 

85 

3359 

61 

2884 

45 

2430 

37   1995 

11 

50 

.7221 

1.3848 

.7490 

1.3351 

.7766 

1.2876 

.8050 

1.2423 

.8342  1.1988 

10 

51 

26 

3840 

95 

3343 

71 

2869 

55 

2415 

46   1981 

9 

52 

30 

3831 

99 

3335 

75 

2861 

59 

2408 

51   1974 

8 

53 

34 

3823 

7504 

3327 

80 

2853 

64 

2401 

56   1967 

7 

54 

39 

3814 

08 

3319 

85 

2846 

69 

2393 

61   1960 

6 

55 

.7243 

1.3806 

.7513 

1.3311 

.7789 

1.2838 

.8074 

1.2386 

.8366  1.1953 

6 

56 

48 

3798 

17 

3303 

94 

2830 

79 

2378 

71   1946 

4 

57 
58 

52 
57 

3789 
3781 

22 
26 

3295 
3287 

99 

2822 

83 
88 

2371 
2364 

76   1939 
81   1932 

3 

2 

7803 

2815 

59 

61 

3772 

31 

3278 

08 

2807 

93 

2356 

86   1925 

1 

60 

.7265 

1.3764 

.7536 

1.3270 

.7813 

1.2799 

.8098 

1.2349 

.8391  1.1918 

0 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot   tan 

'  1     54° 

53° 

52° 

51° 

50^ 

1 

91 


Natural  Sines  and  Cosines 


1 

4C 

r 

41 

o 

42 

o 

43 

o 

44 

o 

/ 

sin 

cos 

sin 

COS 

sin 

cos 

sin 

COS 

sin 

COS 

0 

.6428 

.7660 

.6561 

.7547 

.6691 

.7431 

.6820 

.7314 

.6947 

.7193 

60 

1 

30 

59 

63 

45 

93 

30 

22 

12 

49 

91 

59 

2 

32 

57 

6i 

43 

96 

28 

24 

10 

51 

89 

58 

3 

35 

55 

67 

41 

98 

26 

26 

08 

53 

87 

57 

4 

37 

53 

69 

39 

6700 

24 

28 

06 

55 

85 

56 

6 

.6439 

.7651 

.6572 

.7538 

.6702 

.7422 

.6831 

.7304 

.6957 

.7183 

55 

6 

41 

49 

74 

36 

04 

20 

33 

02 

59 

81 

54 

7 

43 

47 

76 

34 

06 

18 

35 

GO 

61 

79 

53 

8 

46 

45 

78 

32 

09 

16 

37 

7298 

63 

77 

52 

9 

48 

44 

80 

30 

11 

14 

39 

96 

65 

75 

51 

10 

.6450 

.7642 

.6583 

.7528 

.6713 

.7412 

.6841 

.7294 

.6967 

.7173 

50 

11 

52 

40 

85 

26 

15 

10 

43 

92 

70 

71 

49 

12 

55 

38 

87 

24 

17 

08 

45 

90 

72 

69 

48 

13 

57 

36 

89 

22 

19 

06 

48 

88 

^  74 

67 

47 

14 

59 

34 

91 

20 

22 

04 

50 

86 

76 

65 

46 

15 

.6461 

.7632 

.6593 

.7518 

.6724 

.7402 

.6852 

.7284 

.6978 

.7163 

45 

16 

63 

30 

96 

16 

26 

00 

54 

82 

80 

61 

44 

17 

66 

29 

98 

15 

28 

7398 

56 

80 

82 

59 

43 

18 

68 

27 

6600 

13 

30 

96 

58 

78 

84 

57 

42 

19 

70 

25 

02 

11 

32 

94 

60 

76 

86 

55 

41 

20 

.6472 

.7623 

.6604 

.7509 

.6734 

.7392 

.6862 

.7274 

.6988 

.7153 

40 

21 

75 

21 

07 

07 

37 

90 

65 

72 

90 

51 

39 

22 

77 

19 

09 

05 

39 

88 

67 

70 

92 

49 

38 

23 

79 

17 

11 

03 

41 

87 

69 

68 

95 

47 

37 

24 

81 

15 

13 

01 

43 

85 

71 

66 

97 

45 

36 

25 

.6483 

.7613 

.6615 

.7499 

.6745 

.7383 

.6873 

.7264 

.6999 

.7143 

35 

26 

86 

12 

17 

97 

47 

81 

75, 

62 

7001 

41 

34. 

27 

88 

10 

20 

95 

49 

79 

77 

60 

03 

39 

ZZ 

28 

90 

08 

22 

93 

52 

77 

79 

58 

05 

37 

32 

29 

92 

06 

24 

91 

54 

75 

81 

56 

07 

35 

31 

30 

.6494 

.7604 

.6626 

.7490 

.6756 

.7373 

.6884 

.7254 

.7009 

.7133 

30 

31 

97 

02 

28 

88 

58 

71 

86 

52 

11 

30 

29 

32 

99 

00 

31 

86 

60 

69 

88 

50 

13 

28 

28 

2,Z 

6501 

7598 

33 

84 

62 

67 

90 

48 

IS 

26 

27 

34 

03 

96 

35 

82 

64 

65 

92 

46 

17 

24 

26 

35 

.6506 

.7595 

.6637 

.7480 

.6767 

.7363 

.6894 

.7244 

.7019 

.7122 

26 

36 

08 

93 

39 

78 

69 

61 

96 

42 

22 

20 

24 

37 

10 

91 

41 

76 

71 

59 

98 

40 

24 

18 

23 

38 

12 

89 

44 

74 

73 

57 

6900 

38 

26 

16 

22 

39 

14 

87 

46 

72 

75 

55 

03 

36 

28 

14 

21 

40 

.6517 

.7585 

.6648 

.7470 

.6777 

.7353 

.6905 

.7234 

.7030 

.7112 

20 

41 

19 

83 

50 

68 

79 

51 

07 

32 

32 

10 

19 

42 

21 

81 

52 

66 

82 

49 

09 

30 

34 

08 

18 

43 

23 

79 

54 

64 

84 

47 

11 

28 

36 

06 

17 

44 

25 

78 

57 

63 

86 

45 

13 

26 

38 

04 

16 

45 

.6528 

.7576 

.6659 

.7461 

.6788 

.7343 

.6915 

.7224 

.7040 

.7102 

15 

46 

30 

74 

61 

59 

90 

41 

17 

22 

42 

00 

14 

47 

32 

72 

63 

57 

92 

39 

19 

20 

44 

7098 

13 

48 

34 

70 

65 

55 

94 

37 

21 

18 

46 

96 

12 

49 

36 

68 

67 

53 

97 

35 

24 

16 

48 

94 

11 

50 

.6539 

.7566 

.6670 

.7451 

.6799 

.7333 

.6926 

.7214 

.7050 

.7092 

10 

51 

41 

64 

72 

49 

6801 

31 

28 

12 

53 

90 

9 

52 

43 

62 

74 

47 

03 

29 

30 

10 

55 

88 

8 

53 

45 

60 

76 

45 

05 

27 

32 

08 

57 

85 

7 

54 

47 

59 

78 

43 

07 

25 

34 

06 

59 

83 

6 

55 

.6550 

.7557 

.6680 

.7441 

.6809 

.7323 

.6936 

.7203 

.7061 

.7081 

5 

56 

52 

55 

83 

39 

11 

21 

38 

01 

63 

79 

4 

57 

54 

53 

8i 

37 

14 

19 

40 

7199 

65 

77 

3 

58 

56 

51 

87 

35 

16 

18 

42 

97 

67 

75 

2 

59 

58 

49 

89 

33 

18 

16 

44 

95 

69 

73 

1 

60 

.6561 

.7547 

.6691 

.7431 

.6820 

.7314 

.6947 

.7193 

.7071 

.7071 

0 

COS 

sin 

COS 

sin 

cos 

sin 

COS 

sin 

COS 

sin 

/ 

42 

\° 

48 

° 

47 

o 

46 

0 

4£ 

° 

/ 

92 


Natural  Tangents  and  Cotangents 


1 

40°     1 

41°     1 

42° 

43°     1 

44°     1 

1 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

0 

.8391 

1.1918 

.8693 

1.1504 

.9004 

1.1106 

.9325 

1.0724 

.9657 

1.0355 

60 

1 

96 

1910 

98 

1497 

09 

1100 

31 

0717 

63 

0349 

59 

2 

8401 

1903 

8703 

1490 

15 

1093 

36 

0711 

68 

0343 

58 

3 

06 

1896 

08 

1483 

20 

1087 

41 

0705 

74 

0337 

57 

4 

11 

1889 

13 

1477 

25 

1080 

47 

0699 

79 

0331 

56 

5 

.8416 

1.1882 

.8718 

1.1470 

.9030 

1.1074 

.9352 

1.0692 

.9685 

1.0325 

55 

6 

21 

1875 

24 

1463 

36 

1067 

58 

0686 

91 

0319 

54 

7 

26 

1868 

29 

1456 

41 

1061 

63 

0680 

96 

0313 

53 

8 

31 

1861 

34 

1450 

46 

1054 

69 

0674 

9702 

0307 

52 

9 

36 

1854 

39 

1443 

52 

1048 

74 

0668 

08 

0301 

51 

10 

.8441 

1.1847 

.8744 

1.1436 

.9057 

1.1041 

.9380 

1.0661 

.9713 

1.0295 

50 

11 

46 

1840 

49 

1430 

62 

1035 

85 

0655 

19 

0289 

49 

12 

51 

1833 

54 

1423 

67 

1028 

91 

0649 

25 

0283 

48 

13 

56 

1826 

59 

1416 

73 

1022 

96 

0643 

30 

0277 

47 

14 

61 

1819 

65 

1410 

78 

1016 

9402 

0637 

36 

0271 

46 

15 

.8466 

1.1812 

.8770 

1.1403 

.9083 

1.1009 

.9407 

1.0630 

.9742 

1.0265 

45 

16 

71 

1806 

75 

1396 

89 

1003 

13 

0624 

47 

0259 

44 

17 

76 

1799 

80 

1389 

94 

0996 

18 

0618 

53 

0253 

43 

18 

81 

1792 

85 

1383 

99 

0990 

24 

0612 

59 

0247 

42 

19 

86 

1785 

90 

1376 

9105 

0983 

29 

0606 

64 

0241 

41 

20 

.8491 

1.1778 

.8796 

1.1369 

.9110 

1.0977 

.9435 

1.0599 

.9770 

1.0235 

40 

21 

96 

1771 

8801 

1363 

15 

0971 

40 

0593 

76 

0230 

39 

22 

8501 

1764 

06 

1356 

21 

0964 

46 

0587 

81 

0224 

38 

23 

06 

1757 

11 

1349 

26 

0958 

51 

0581 

87 

0218 

37 

24 

11 

1750 

16 

1343 

31 

0951 

57 

0575 

93 

0212 

36 

25 

.8516 

1.1743 

.8821 

1.1336 

.9137 

1.0945 

.9462 

1.0569 

.9798 

1.0206 

35 

26 

21 

1736 

27 

1329 

42 

0939 

68 

0562 

9804 

0200 

34 

27 

26 

1729 

32 

1323 

47 

0932 

73 

0556 

10 

0194 

33 

28 

31 

1722 

37 

1316 

53 

0926 

79 

0550 

16 

0188 

32 

29 

36 

1715 

42 

1310 

58 

0919 

84 

0544 

21 

0182 

31 

30 

.8541 

1.1708 

.8847 

1.1303 

.9163 

1.0913 

-.9490 

1.0538 

.9827 

1.0176 

30 

31 

46 

1702 

52 

1296 

69 

0907 

95 

0532 

33 

0170 

29 

32 

51 

1695 

58 

1290 

74 

0900 

9501 

0526 

38 

0164 

28 

33 

56 

1688 

63 

1283 

79 

0894 

06 

0519 

44 

0158 

27 

34 

61 

1681 

68 

1276 

85 

0888 

12 

0513 

50 

0152 

26 

35 

.8566 

1.1674 

.8873 

1.1270 

.9190 

1.0881 

.9517 

1.0507 

.9856 

1.0147 

25 

36 

71 

1667 

78 

1263 

95 

0875 

23 

0501 

61 

0141 

24 

37 

76 

1660 

84 

1257 

9201 

0869 

28 

0495 

67 

0135 

23 

38 

81 

1653 

89 

1250 

06 

0862 

34 

0489 

73 

0129 

22 

39 

86 

1647 

94 

1243 

12 

0856 

40 

0483 

79 

0123 

21 

40 

.8591 

1.1640 

.8899 

1.1237 

.9217 

1.0850 

.9545 

1.0477 

.9884 

1.0117 

20 

41 

96 

1633 

8904 

1230 

22 

0843 

51 

0470 

90 

0111 

19 

42 

8601 

1626 

10 

1224 

28 

0837 

56 

0464 

96 

0105 

18 

43 

06 

1619 

15 

1217 

33 

0831 

62 

0458 

9902 

0099 

17 

44 

11 

1612 

20 

1211 

39 

0824 

67 

0452 

07 

0094 

16 

45 

.8617 

1.1606 

.8925 

1.1204 

.9244 

1.0818 

.9573 

1.0446 

.9913 

1.0088 

15 

46 

22 

1599 

31 

1197 

49 

0812 

78 

0440 

19 

0082 

14 

47 

27 

1592 

36 

1191 

55 

0805 

84 

0434 

2i 

0076 

13 

48 

32 

1585 

41 

1184 

60 

0799 

90 

0428 

30 

0070 

12 

49 

37 

1578 

46 

1178 

66 

0793 

95 

0422 

36 

0064 

11 

60 

.8642 

1.1571 

.8952 

1.1171 

.9271 

1.0786 

.9601 

1.0416 

.9942 

1.0058 

10 

51 

47 

1565 

57 

1165 

76 

0780 

06 

0410 

48 

0052 

9 

52 

52 

1558 

62 

1158 

82 

0774 

12 

0404 

54 

0047 

8 

53 

57 

1551 

67 

1152 

87 

0768 

18 

[0398 

59 

0041 

7 

54 

62 

1544 

72 

1145 

93 

0761 

23 

0392 

65 

0035 

6 

55 

.8667 

1.1538 

.8978 

1.1139 

.9298 

1.0755 

.9629 

1.0385 

.9971 

1.0029 

5 

56 

72 

1531 

83 

1132 

9303 

0749 

34 

0379 

77 

0023 

4 

57 

78 

1524 

88 

1126 

09 

0742 

40 

0373 

83 

0017 

3 

58 

83 

1517 

94 

1119 

14 

0736 

46 

0367 

88 

0012 

2 

59 

88 

1510 

99 

1113 

20 

0730 

51 

0361 

94 

0006 

1 

60 

.8693 

1.1504 

.90(H 

1.1106 

.9325 

1.0724 

.9657 

1.0355 

1.0000 

1.0000 

0 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

1 

1 

49° 

48° 

47° 

46° 

45° 

93 


TABLE  IV 


raSCELLAJVEOUS  TABLES 


1.  Radian  Measure,  0°  to  180°. 

2.  Natural  Logarithms: 

(A)  Numbers  from  1  to  200. 

(B)  Numbers  from  1  to  9.9. 

8.    The  Hyperbolic  Functions,  Sinhx,  Coshx. 


95 


TABLE  IV.        1.     Eadian  Measure,  0°  to  180°,  Radius  =  1 


DEGREES 

MINUTES 

SECONDS    1 

0° 

1 

2 
3 

0.00000  00 
0.01745  33 
0.03490  66 
0.05235  99 

60° 

61 
62 
63 

1.04719  76 

120° 

121 
122 
123 

2.09439  51 
2.11184  84 
2.12930  17 
2.14675  50 

0' 

1 

2 
3 

0.00000  00 

0" 

1 
2 
3 

0.00000  00 

1.06465  08 
1.08210  41 
1.09955  74 

0.00029  09 
0.00058  18 
0.0Q0S7  27 

0.00000  48 
0.00000  97 
0.00001  45 

4 
5 
6 

0.06981  32 
0.08726  65 
0.1047198 

64 
65 
66 

1.11701  07 
1.13446  40 
1.15191  73 

124 
125 
126 

2.16420  83 
2.18166  16 
2.19911  49 

4 
5 
6 

0.00116  36 
0.00145  44 
0.00174  53 

4 
5 
6 

0.00001  94 
0.00002  42 
0.00002  91 

7 
8 
9 

10 
11 
12 
13 

0.12217  30 
0.13962  63 
0.15707  96 

67 
68 
69 
70 

71 

72 
73 

1.16937  06 
1.18682  39 
1.20427  72 

127 
128 
129 
130 

131 

132 
133 

2.21656  82 
2.23402  14 
2.25147  47 
2.26892  80 

7 
8 
9 

10 
11 
12 
13 

0.00203  62 
0.00232  71 
0.00261  80 

7 
8 
9 
10 
11 
12 
13 

0.00003  39 
0.00003  88 
0.00004  36 

0.17453  29 

1.22173  05 

0.00290  89 

0.00004  85 

0.19198  62 
0.20943  95 
0.226S9  28 

1.23918  38 
1.25663  71 
1.27409  04 

2.28638  13 
2.30383  46 
2.32128  79 

0.00319  98 
0.00349  07 
0.00378  15 

0.00005  33 
0.00005  82 
0.00006  30 

14 
15 
16 

0.24434  61 
0.26179  94 
0.27925  27 

74 
75 
76 

1.29154  36 
1.30899  69 
1.32645  02 

134 
135 
136 

2.33874  12 
2.35619  45 
2.37364  78 

14 
15 
16 

0.00407  24 
0.00436  33 
0.00465  42 

14 
15 
16 

0.00006  79 
0.00007  27 
0.00007  76 

17 
18 
19 
20 

21 
22 
23 

0.29670  60 
0.31415  93 
0.33161  26 

77 
78 
79 
80 
81 
82 
83 

1.34390  35 
1.36135  68 
1.37881  01 

137 
138 
139 
140 
141 
142 
143 

2.39110  11 
2.40855  44 
2.42600  77 

17 
18 
19 
20 

21 
22 
23 

0.00494  51 
0.00523  60 
0.00552  69 

17 
18 
19 
20 
21 
22 
23 

0.00008  24 
0.00008  73 
0.00009  21 

0.34906  59 

1.39626  34 

2.44346  10 

0.00581  78 

0.00009  70 

0.36651  91 
0.38397  24 
0.40142  57 

1.41371  67 
1.43117  00 
1.44862  33 

2.46091  42 
2.47836  75 
2.49582  08 

0.00610  87 
0.00639  95 
0.00669  04 

0.00010  18 
0.00010  67 
0.00011  15 

24 
25 
26 

0.41887  90 
0.43633  23 
0.45378  56 

84 
85 
86 

1.46607  66 
1.48352  99 
1.50098  32 

144 
145 
146 

2.51327  41 
2.53072  74 
2.54818  07 

24 
25 
26 

0.00698  13 
0.00727  22 
0.00756  31 

24 
25 
26 

0.00011  64 
0.00012  12 
0.00012  61 

27 
28 
29 
30 
31 
32 
33 

0.47123  89 
0.48869  22 
0.50614  55 

87 
88 
89 
90 
91 
92 
93 

1.51843  64 
1.53588  97 
1.55334  30 

147 
148 
149 
150 
151 
152 
153 

2.  .56563  40 
2.58308  73 
2.60054  06 

27 
28 
29 
30 
31 
32 
33 

0.00785  40 
0.00814  49 
0.00843  58 

27 
28 
29 
30 
31 
32 
33 

0.00013  09 
0.00013  57 
0.00014  06 
0:00014  54 

0.52359  88 

1.57079  63 

2.61799  39 

0.00872  66 

0.54105  21 
0.55850  54 
0.57595  87 

1.58824  96 
1.60570  29 
1.62315  62 

2.63544  72 
2.65290  05 
2.67035  38 

0.00901  75 
0.00930  84 
0.00959  93 

0.00015  03 
0.00015  51 
0.00016  00 

34 
35 
36 

0.59341  19 
0.61086  52 
0.62831  85 

94 
95 
96 

1.64060  95 
1.65806  28 
1.67551  61 

154 
155 
156 

2.68780  70 
2.70526  03 
2.72271  36 

34 
35 
36 

0.00989  02 
0.01018  11 
0.01047  20 

34 
35 
36 

0.00016  48 
0.00016  97 
0.00017  45 

37 
38 
39 
40 
41 
42 
43 

0.64577  18 
0.66322  51 
0.68067  84 

97 
98 
99 
100 

101 
102 
103 

1.69296  94 
1.71042  27 
1.72787  60 

157 
158 
159 
160 
161 
162 
163 

2.74016  69 
2.75762  02 
2.77507  35 

37 
38 
39 
40 
41 
42 
43 

0.01076  29 
0.01105  38 
0.01134  46 

37 
38 
39 
40 
41 
42 
43 

0.00017  94 
0.00018  42 
0.00018  91 

0.69813  17 

1.74532  93 

2.79252  68 

0.01163  55 

0.00019  39 

0.71558  50 
0.73303  83 
0.75049  16 

1.76278  25 
1.78023  58 
1.79768  91 

2.80998  01 
2.82743  34 
2.84488  67 

0.01192  64 
0.01221  73 
0.01250  82 

0.00019  88 
0.00020  36 
0.00020  85 

44 
45 
46 

0.76794  49 
0.78539  82 
0.80285  15 

104 
105 
106 

1.81514  24 
1.83259  57 
1.85004  90 

164 
165 
166 

2.86234  00 
2.87979  33 
2.89724  66 

44 
45 
46 

0.01279  91 
0.01309  00 
0.01338  09 

44 
45 
46 

0.00021  33 
0.00021  82 
0.00022  30 

47 
48 
49 
50 

51 

52 
53 

0.82030  47 
0.83775  80 
0.85521  13 

107 
108 
109 
110 
111 
112 
113 

1.86750  23 
1.88495  56 
1.90240  89 

167 
168 
169 
170 
171 
172 
173 

2.91469  99 
2.93215  31 
2.94960  64 

47 
48 
49 
50 

51 

52 
53 

0.01367  17 
0.01396  26 
0.01425  35 

47' 
48 
49 
50 

51 

52 
53 

0.00022  79 
0.00023  27 
0.00023  76 

0.87266  46 

1.91986  22 

2.96705  97 

0.01454  44 

0.00024  24 

0.89011  79 
0.90757  12 
0.92502  45 

1.93731  55 
1.95476  88 
1.97222  21 

2.98451  30 
3.00196  63 
3.01941  96 

0.01483  53 
0.01512  62 
0.01541  71 

0.00024  73 
0.00025  21 
0.00025  70 

54 
55 
56 

0.94247  78 
0.95993  11 
0.97738  44 

114 
115 
116 

1.98967  53 
2.00712  86 
2.02458  19 

174 
175 
176 

3.03687  29 
3.05432  62 
3.07177  95 

54 
55 
56 

0.01570  80 
0.01599  89 
0.01628  97 

54 
55 
56 

0.00026  IS 
0.00026  66 
0.00027  15 

57 
58 
59 
60 

0.99483  77 
1.01229  10 
1.02974  43 

117 
118 
119 
120 

2.04203  52 
2.05948  85 
2.07694  18 

177 
178 
179 
180 

3.08923  28 
3.10668  61 
3.12413  94 

57 
58 
59 
60 

0.01658  06 
0.01687  15 
0.01716  24 

57 
58 
59 
60 

0.00027  63 
0.00028  12 
0.00028  60 

1.04719  76 

2.09439  51 

3.14159  27 

0.01745  33 

0.00029  09 

DEGBEES 

J 

fINUTES 

SECONDS    1 

96 


TABLE   IV 


2.     Natural   Logarithms 


A. 

THE  NATURAL  LOGARITHMS 

INTEGERS 

FROM  1  TO  200 

Base  6  =  2.7182818284... 

Conversion  Laws : 

logj^  6  =  0.4342944819... 
log,  10  =2.3025850929... 

log^7r  =  1.1447298858... 

log,iV^  =  log,10  xlogioiV. 

N  1   loge 

N 

loge 

N 

loge 

N     loge 

N 

loge 

0 

1 

2 
3 

—  CO 

40 

41 
42 
43 

3.68  888 

80 

81 
82 
83 

4.3 

8  203 

120 

121 
122 
123 

4.78  749 

160 

161 
162 
163 

5.07  517 

0.00  000 
0.69  315 
1.09  861 

3.71  357 
3.73  767 
3.76  120 

4.3 
4.-1 
4.^ 

9  445 

0  672 

1  884 

4  79  579 

4.80  402 

4.81  218 

5.08  140 

5.08  760 

5.09  375 

4 
5 
6 

1.38  629 
1.60  944 
1.79  176 

44 
45 
46 

3.78  419 
3.80  666 
3.82  864 

84 
85 
86 

4.4 
4.4 
4.4 

3  082 

4  265 

5  435 

124 
125 
126 

4.82  028 

4.82  831 

4.83  628 

164 
165 
166 

5.09  987 

5.10  595 

5.11  199 

7 
8 
9 

10 

11 
12 
13 

1.94  591 
2.07  944 
2.19  722 

47 
48 
49 

50 

51 
52 
53 

3.85  015 
3.87  120 
3.89  182 

87 
88 
89 

90 

91 
92 
93 

4.4 
4.4 
4.4 

6  591 

7  7.34 

8  864 

127 
128 
129 

130 

131 
132 
133 

4.84  419 

4.85  203 
4.85  981 

167 
168 
169 

170 

171 
172 
173 

5.11  799 

5.12  396 
5.12  990 

2.30  259 

2.39  790 
2.48  491 
2.56  495 

3.91  202 

4.4 

9  981 

4.86  753 

5.13  580 

3.93  183 
3.95  124 
3.97  029 

4.5 
4.5 
4.5 

1  086 

2  179 

3  260 

4.87  520 
4.SS  280 
4.89  035 

5.14  166 

5.14  749 

5.15  329 

U 
15 
16 

2.63  906 
2.70  805 
2.77  259 

54 
55 
56 

3.98  898 
4.00  733 
4.02  535 

94 
95 
96 

4.5 
4.5 
4.5 

4  329 

5  388 

6  435 

134 
135 
136 

4.89  784 

4.90  527 

4.91  265 

174 
175 
176 

5.15  906 

5.16  479 

5.17  048 

17 
18 
19 

20 

21 
22 
23 

2.83  321 
2.89  037 
2.94  444 

57 
58 
59 

60 

61 
62 
63 

4.04  305 

4.06  044 

4.07  754 

97 
98 
99 

100 

101 
102 
103 

4.5 
4.5 
4.5 

7  471 

8  497 

9  512 

137 
138 
139 

140 

141 
142 
143 

4.91  998 

4.92  725 

4.93  447 

177 
178 
179 

180 

181 
182 
183 

5.17  615 

5.18  178 
5.18  739 

2.99  573 

4.09  434 

4.6 

0  517 

4.94  164 

5.19  296 

3.04  452 
3.09  104 
3.13  549 

4.11  087 

4.12  713 
4.14  313 

4.6 
4.6 

4.6 

1  512 

2  497 

3  473 

4.94  876 

4.95  583 

4.96  284 

5.19  850 

5.20  401 
5.20  949 

24 
25 
26 

3.17  805 
3.21  888 
3.25  810 

64 
65 
66 

4.15  888 

4.17  439 

4.18  965 

104 
105 
106 

4.6 
4.6 
4.6 

4  439 

5  396 

6  344 

144 
145 
146 

4.96  981 

4.97  673 

4.98  361 

184 
185 
186 

5.21  494 

5.22  036 
5.22  575 

27 
28 
29 

30 

31 
32 
33 

3.29  584 
3.33  220 
3.36  730 

67 
68 
69 

70 

71 

72 
73 

4.20  469 

4.21  951 
4.23  411 

107 
108 
109 

110 

111 
112 
113 

4.6 
4.6 
4.6 

7  283 

8  213 

9  135 

147 
148 
149 

150 

151 
152 
153 

4.99  043 
4.99  721 
5.00  395 

187 
188 
189 

190 

191 
192 
193 

5.23  111 

5.23  644 

5.24  175 

3.40  120 

4.24  850 

4.7 

0  048 

5.01  064 

5.24  702 

3.43  399 
3.46  574 
3.49  651 

4.26  268 

4.27  667 
4.29  046 

4.7 
4.7 
4.7 

0  953 

1  850 

2  739 

5.01  728 

5.02  388 

5.03  044 

5.25  227 

5.25  750 

5.26  269 

34 
35 
36 

3.52  636 

3.55  535 
3.58  352 

74 
75 
76 

4.30  407 

4.31  749 
4.33  073 

114 
115 
116 

4.7 
4.7 

4.7 

3  620 

4  493 

5  359 

L54 
155 
156 

5.03  695 

5.04  343 
5.04  986 

194 
195 
196 

5.26  786 

5.27  300 
5.27  811 

37 
38 
39 

40 

3.61  092 
3.63  759 
3.66  356 

77 
78 
79 

80 

4.34  381 

4.35  671 

4.36  945 

117 
118 
119 

120 

4.7 
4.7 
4.7 

6  217 

7  068 
7  912 

157 
158 
159 

160 

5.05  625 

5.06  260 
5.06  890 

197 
198 
199 

200 

5.28  320 

5.28  827 

5.29  330 

3.68  888 

4.38  203 

4.7 

8  749 

5.07  517 

5.29  832 

97 


TABLE   IV. 


Natural    Locakithms 


B. 

NATURAL  LOGARITHMS  1 

TO  9.9 

The  following  table  shows  the  Natural  or  Napierian  log- 

arithms, for  each  tenth,  of  numbers  1  to  9.9.     Interpola- 

tion may  be  made  for  hundredths.    'The  logarithms  of 

numbers  larger  than  9.9  may  be  found  as  shown  in  the 

following  illustration:     Let  us  find  the  log ^450: 

log^  450  =  log,  (4.5  xl(F)  =  log,  4.5  +  2  log,  10 

=  1.5041  +  2  (2.3026)  =  6.1093. 

XO.  1    lOge 

No. 

lOge 

>'0.  1     loge 

No.      loge 

No.     log«^ 

No.       lOffe 

1.0 

0.0000 

~ 

0.9163 

4.0 

1.3863 

5.5 

1.7048 

7.0 

1.9459 

S.5  1  2.1401 

1.1 

.0953 

2.6 

.9555 

4.1 

.4110 

5.6 

.7228 

7.1 

.9601 

8.6 

.1518 

1.2 

.1823 

2.7 

.9933 

4.2 

.4351 

5.7 

.7405 

7.2 

.9741 

8.7 

.1633 

1.3 

.2624 

2.8 

1.0296 

4.3 

.4586 

5.8 

.7579 

7.3 

.9879 

8.8 

.1748 

1.4 

.3365 

2.9 

.0647 

4.4 

.4816 

5.9 

.7750 

7.4 

2.0015 

8.9 

.1861 

1.5 

0.4055 

3.0 

1.0986 

4.5 

1.5041 

6.0 

1.7917 

7.5 

2.0149 

9.0 

2.1972 

1.6 

.4700 

3.1 

.1314 

4.6 

.5261 

6.1 

.8083 

7.6 

.0282 

9.1 

.2083 

1.7 

.5306 

3.2 

.1632 

4.7 

.5476 

6.2 

.8246 

7.7 

.(H12 

9.2 

.2192 

1.8 

.5878 

3.3 

.1939 

4.8 

.5686 

6.3 

.8406 

7.8 

.0541 

9.3 

.2300 

1.9 

.6419 

3.4 

.2238 

4.9 

.5892 

6.4 

.8563 

7.9 

.0669 

9.4 

.2407 

2.0 

0.6932 

3.5 

1.2528 

5.0 

1.6094 

6.5 

1.8718 

8.0 

2.0794 

9.5 

2.2513 

2.1 

.7419 

3.6 

.2809 

5.1 

.6292 

6.6 

.8871 

8.1 

.0919 

9.6 

.2618 

2.2 

.7885 

3.7 

.3083 

5.2 

.6487 

6.7 

.9021 

8.2 

.1041 

9.7 

.2721 

2.3 

.8329 

3.8 

.3350 

5.3 

.6677 

6.8 

.9169 

8.3 

.1163 

9.8 

.2824 

2.4 

.8755 

3.9 

.3610 

5.4 

.6864 

6.9 

.9315 

8.4 

.1282 

9.9 

.2925 

98 


TABLE   lY 
Hyperbolic  Si>-hx  =  — ~^    ,  Coshx  =  — ^tf — . 


vT 

sinh  J- 

cmhx 

j: 

!«inh  J- 

cosh  X 

0.00 

0.0000 

1.0000 

.   0.50 

0.i>211 

1.1276 

.01 

.0100 

1.0000 

.51 

.5324 

1.1329 

.02 

.0200 

1.0002 

!    .52 

.5438 

1.1383 

.03 

.0300 

l.OOOf 

% 

.5552 

1.1438 

.(H 

.otoo 

1.0008 

.5666 

1.1494 

.05 

.0500 

1.0013 

.55 

.5782 

1.1551 

.06 

.0600 

1.0018 

.56 

.5897 

1.1609 

.07 

.0701 

1.0025 

.57 

.6014 

1.1669 

.OS 

.OSOl 

1.0032 

.58 

.6131 

1.1730 

.09 

.0901 

1.0041 

.59 

.6248 

1.1792 

.10 

.1002 

1.0050 

.60 

.6367 

1.1855 

.11 

.1102 

1.0061 

.61 

.6485 

1.1919 

.12 

.1203 

1.0072 

.62 

.6605 

1.1984 

.13 

.13(H 

1.0085 

.63 

.6725 

1.2051 

.14 

.1405 

1.0098 

.64 

.6846 

1.2119 

.15 

.1506 

1.0113 

.65 

.6967 

1.2188 

.16 

.1607 

1.0128 

.66 

.7090 

1.2258 

.17 

.1708 

1.0145 

.67 

.7213 

1.2330 

.IS 

.1810 

1.0162 

.68 

.7336 

1.2402 

.19 

.1911 

1.0181 

.69 

.7461 

1.2476 

.20 

.2013 

1.0201 

.70 

.7586 

1.2552 

.21 

.2115 

1.0221 

.71 

.7712 

\.?f>?S 

.22 

.2218 

1.0243 

.72 

.7838 

1.2706 

.23 

.2320 

1.0266 

.73 

.7%6 

1.2785 

.24 

.2423 

1.0289 

.74 

.aib9+ 

1.2865 

.25 

.2526 

1.0314 

.75 

.8223 

1.2947 

.26 

.2629 

1.0340 

.76 

.8353 

1.3030 

.27 

.2733 

1.0367 

.77 

.8484 

1.3114 

.2S 

.2837 

1.0395 

.78 

.8615 

1.3199 

.29 

.2941 

1.0423 

.79 

.8748 

1.3286 

.30 

.3045 

l.(H53 

.80 

.8881 

1.3374 

.31 

.3150 

1.0484 

.81 

.9015 

1.3464 

.32 

.3255 

1.0516 

.82 

.9150 

1.3555 

.33 

.3360 

1.0549 

.83 

.9286 

1.3647 

.34 

.3466 

1.05&t 

.84 

.9423 

1.3740 

.35 

.3572 

1.0619 

.85 

.9561 

1.3835 

.36 

.3678 

1.0655 

.86 

.9700 

1.3932 

.37 

.3785 

1.0692 

.87 

.9840 

1.4029 

.38 

.3892 

1.0731 

.88 

.9981 

1.4128 

.39 

.4000 

1.0770 

.89 

1.0122 

1.4229 

.40 

.4108 

1.0811 

.90 

1.0265 

1.4331 

.41 

.4216 

1.0SS2 

.91 

1.0409 

1.4434 

.42 

.4325 

1.0S95 

.92 

1.0554 

1.4539 

.43 

.4434 

1.0939 

.93 

1.0700 

1.4645 

.44 

.4543 

1.0984 

.94 

1.0847 

1.4753 

.45 

.4653 

1.1030   j 

.95 

1.0995 

1.4862 

.46 

.4764 

1.1077 

.96 

1.1144 

1.4973 

.47 

.4875 

1.1125 

.97 

1.1294 

1.5085 

.48 

.4986 

1.1174 

.98 

1.1446 

1.5199 

.49 

.5098 

1.1225 

.99 

1.1598 

1.5314 

0.50 

0.5211 

1.1276   i 

1.00 

1.1752 

1.5431 

•ti» 


First  Course  in 
Differential  and  Integral  Calculus 

By  WILLIAM  F.  OSGOOD,  Ph.D. 

Professor  of  Mathematics  in  Harvard  University 


Revised  Edition.     Cloth,  xv  +  462  pages,  $2.00  net 


The  treatment  of  this  calculus  by  Professor  Osgood  is  based  on 
the  courses  he  has  given  in  Harvard  College  for  a  number  of 
years.  The  chief  characteristics  of  the  treatment  are  the  close 
touch  between  the  calculus  and  those  problems  of  physics,  including 
geometry,  to  which  it  owed  its  origin ;  and  the  simplicity  and 
directness  with  which  the  principles  of  the  calculus  are  set  forth. 
It  is  important  that  the  formal  side  of  the  calculus  should  be 
thoroughly  taught  in  a  first  course,  and  great  stress  has  been  laid 
on  this  side.  But  nowhere  do  the  ideas  that  underlie  the  calculus 
come  out  more  clearly  than  in  its  applications  to  curve  tracing 
and  the  study  of  curves  and  surfaces,  in  definite  integrals,  with 
their  varied  applications  to  physics  and  geometry,  and  in 
mechanics.  For  this  reason  these  subjects  have  been  taken  up 
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ciple developed  is  followed  by  a  number  of  applications.  In 
many  cases  these  are  illustrated,  and  they  all  deal  with  matters 
that  directly  concern  the  engineer.  It  is  believed  that  the 
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present  volume.  Accordingly,  the  title  "  Applied  Mechanics  for 
Engineers  "  has  been  given  to  the  book.  The  book  is  intended 
as  a  text-book  for  engineering  students  of  the  Junior  year.  The 
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semester.  In  some  chapters  more  material  is  presented  than  can 
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